Data-driven representation and clustering discretization method and system for design optimization and/or performance prediction of material systems and applications of same

ABSTRACT

A method and system for design optimization and/or performance prediction of a material system includes generating a representation of the material system at a number of scales, the representation at a scale comprising microstructure volume elements (MVE) of building blocks of the material system at said scale; providing inputs to the MVEs; collecting data of response fields of the MVE computed from a material model of the material system over a predefined set of material properties and boundary conditions; applying machine learning to the collected data to generate clusters; computing an interaction tensor of interactions of each cluster with each of the other clusters; and solving an governing partial differential equation using the generated clusters and the computed interactions to result in a response prediction usable in an iterative scheme in a multiscale model for the material system. The performance of each scale can be predicted for design optimization.

CROSS-REFERENCE TO RELATED PATENT APPLICATION

This application claims priority to and the benefit of, pursuant to 35U.S.C. § 119(e), of U.S. provisional patent application Ser. No.62/731,381, filed Sep. 14, 2018, entitled “MULTISCALE MODELING PLATFORMAND APPLICATIONS OF SAME”, by Wing Kam Liu, Jiaying Gao, Cheng Yu andOrion L. Kafka, which is incorporated herein by reference in itsentirety.

This application is related to a co-pending U.S. patent application,entitled “INTEGRATED PROCESS-STRUCTURE-PROPERTY MODELING FRAMEWORKS ANDMETHODS FOR DESIGN OPTIMIZATION AND/OR PERFORMANCE PREDICTION OFMATERIAL SYSTEMS AND APPLICATIONS OF SAME”, by Wing Kam Liu, JiayingGao, Cheng Yu, and Orion L. Kafka, with Attorney Docket No.0116936.152US2, filed on the same day that this application is filed,and with the same assignee as that of this application, which isincorporated herein by reference in its entirety.

FIELD OF THE INVENTION

The invention relates generally to materials, and more particularly, toa specific method and system to aggregate dissimilar material geometry,properties, and interactions to predict combined properties andperformance and applications of the same.

BACKGROUND OF THE INVENTION

The background description provided herein is for the purpose ofgenerally presenting the context of the invention. The subject matterdiscussed in the background of the invention section should not beassumed to be prior art merely as a result of its mention in thebackground of the invention section. Similarly, a problem mentioned inthe background of the invention section or associated with the subjectmatter of the background of the invention section should not be assumedto have been previously recognized in the prior art. The subject matterin the background of the invention section merely represents differentapproaches, which in and of themselves may also be inventions. Work ofthe presently named inventors, to the extent it is described in thebackground of the invention section, as well as aspects of thedescription that may not otherwise qualify as prior art at the time offiling, are neither expressly nor impliedly admitted as prior artagainst the invention.

Microstructured materials are aggregates of individual components.Although many approximations exist, high-fidelity modeling of a complexmaterial typically involves building a numerical model of itsmicrostructure. In the field of computational mechanics, virtual testsfor the microstructure responses are often carried out usinghigh-fidelity numerical methods, such as the Finite Element Method(FEM), Finite Volume Method (FVM), Fast Fourier Transform (FFT), andmesh free. This process usually requires building an explicit mesh insoftware for the microstructure, where the mesh is the virtualrepresentation of the microstructure. This is typically a spatialdecomposition where the elements that make up the discrete units of thedecomposition follow specific rules (e.g., non-negative Jacobian forFEM, fixed spacing grid in FFT). Although there are differences, thesespatial discretizations are simply referred as a “meshes.” In order tocapture all the details of the microstructure, the mesh usually requiresvery fine resolution. This discretization of the material based on spaceis similar to the concept of a Riemann integral, as illustrated in FIG.2. Typically, a fine mesh of very simple fiber composite materials maycontain millions of elements.

Once the mesh for a microstructure is constructed, loads can be appliedto the mesh to perform virtual testing. Various information can beobtained from this virtual testing, such as elastic properties of thematerial, effective strength of the microstructure, and local stressdistributions to study possible microstructure failure. However,resolving a fine mesh requires significant computational resource, suchas High Performance Cluster computing or GPU computing, and a longexecution time for the finite element software.

For a Microstructure Volume Element (a representation of the complexmaterial microstructure), or MVE in short, at any given scale, the localand homogenized behavior at the current scale might be computed as afunction of the homogenized behavior at subscales using traditionalnumerical methods, such as finite element analysis. However, thecomputation time is proportional to the total number of elements acrossall scales. Each added scale compounds the difficulty. Therefore, thecomputation time can be high for even a simple 2-scale model if the MVEcontains many elements—a typical MVE might contain O(1e6) elements, andthis would have to be solved for each element at the larger scale. Thismeans cost is very high. When done with finite elements for two scales,this is sometimes called an “FE²”. Because of the cost, there is limitedpractical application of such schemes, and they have seen little use.

Therefore, a heretofore unaddressed need exists in the art to addressthe aforementioned deficiencies and inadequacies.

SUMMARY OF THE INVENTION

In order to counter the time-consuming microstructure simulation, one ofthe objectives of this invention is to develop a data-driven domaindecomposition approach that is suitable to accelerating the numericalsimulation of the microstructure responses.

In one aspect, the invention relates to a method for design optimizationand/or performance prediction of a material system. In one embodiment,the method includes generating a representation of the material systemat a number of scales, wherein the representation at a scale comprisesmicrostructure volume elements (MVE) that are building blocks of thematerial system at said scale; collecting data of response fields of theMVE computed from a material model of the material system over apredefined set of material properties and boundary conditions; applyingmachine learning to the collected data of response fields to generateclusters that minimize a distance between points in a nominal responsespace within each cluster; computing an interaction tensor ofinteractions of each cluster with each of the other clusters;manipulating the governing partial differential equation (PDE) usingGreen's function to form a generalized Lippmann-Schwinger integralequation; and solving the integral equation using the generated clustersand the computed interactions to result in a response prediction that isusable for the design optimization and/or performance prediction of thematerial system.

In one embodiment, the method further comprises passing the resultedresponse prediction to a next coarser scale as an overall response ofthat building block, and iterating the process until a final scale isreached.

In one embodiment, the building blocks are defined by materialproperties and structural descriptors obtained by modeling orexperimental observations and encoded in a domain decomposition ofstructures for identifying locations and properties of each phase withinthe building blocks.

In one embodiment, the structural descriptors comprise characteristiclength and geometry.

In one embodiment, the boundary conditions are chosen to satisfy theHill-Mandel condition.

In one embodiment, the collected data of response fields comprise astrain concentration tensor, a deformation concentration tensor, stresstensor (e.g., PK-I stress, Cauchy stress), plastic strain tensor,thermal gradient, or the like.

In one embodiment, the machine learning comprises unsupervised machinelearning and/or supervised machine learning.

In one embodiment, the machine learning is performed with aself-organizing mapping (SOM) method, a k-means clustering method, orthe like.

In one embodiment, the clusters are generated by marking all materialpoints that have the same response field within the representation ofthe material system with a unique ID and grouping material points withthe same ID.

In one embodiment, the generated clusters are a reduced representationof the material system, which reduces the number of degrees of freedomrequired to represent the material system.

In one embodiment, the generated clusters are a reduced order MVE of thematerial system.

In one embodiment, the computed interaction tensor is for all pairs ofthe clusters.

In one embodiment, said computing of the interaction tensor is performedwith fast Fourier transform (FFT), numerical integration, or finiteelement method (FEM).

In one embodiment, the PDE is reformulated as a Lippmann-Schwingerequation. In one embodiment, said solving of the PDE is performed witharbitrary boundary conditions and/or material properties.

In one embodiment, the collected data of response fields, the generatedclusters, and/or the computed interaction tensor are saved in one ormore material system databases.

In one embodiment, said solving of the PDE is performed in real-time byaccessing the one or more material system databases for the generatedclusters and the computed interactions.

In another aspect, the invention relates to a method for designoptimization and/or performance prediction of a material system. In oneembodiment, the method includes performing an offline data compression,wherein original MVE of building blocks of the material system arecompressed into clusters, and an interaction tensor of interactions ofeach cluster with each of the other clusters is computed; and solving angoverning PDE using the clusters and the computed interactions to resultin a response prediction that is usable for the design optimizationand/or performance prediction of the material system.

In one embodiment, the method further includes passing the resultingresponse prediction to a next coarser scale as an overall response ofthat building block, and iterating the process until a final scale isreached.

In one embodiment, the building blocks are defined by materialproperties and structural descriptors obtained by modeling orexperimental observations and encoded in a domain decomposition ofstructures for identifying locations and properties of each phase withinthe building blocks.

In one embodiment, the structural descriptors comprise characteristiclength and geometry.

In one embodiment, when more than one scale is involved with the reducedorder response prediction, the method is named MultiresolutionClustering Analysis (MCA).

In one embodiment, the boundary conditions are chosen to satisfy theHill-Mandel condition.

In one embodiment, said performing the offline data compressioncomprises collecting data of response fields of the MVE computed from amaterial model of the material system over a predefined set of materialproperties and boundary conditions; applying machine learning to thecollected data of response fields to generate clusters that minimize adistance between points in a nominal response space within each cluster;and computing the interaction tensor is for all pairs of the clusters.

In one embodiment, the collected data of response fields comprise astrain concentration tensor, a deformation concentration tensor, stresstensor (e.g., PK-I stress, Cauchy stress), plastic strain tensor,thermal gradient, or the like.

In one embodiment, the machine learning comprises unsupervised machinelearning and/or supervised machine learning.

In one embodiment, the machine learning is performed with an SOM method,a k-means clustering method, or the like.

In one embodiment, the clusters are generated by marking all materialpoints having the same response field within the representation of thematerial system with a unique ID and grouping material points with thesame ID.

In one embodiment, the clusters are a reduced representation of thematerial system, which reduces the number of degrees of freedom requiredto represent the material system.

In one embodiment, the clusters are a reduced order MVE of the materialsystem.

In one embodiment, said computing the interaction tensor is performedwith FFT, numerical integration, or FEM.

In one embodiment, the PDE is a Lippmann-Schwinger equation. In oneembodiment, said solving the PDE is performed with arbitrary boundaryconditions and material properties.

In one embodiment, the collected data of response fields, the generatedclusters, and/or the computed interaction tensor are saved in one ormore material system databases.

In one embodiment, said solving the PDE is performed with onlineaccessing the one or more material system databases for the generatedclusters and the computed interactions.

In yet another aspect, the invention relates to a non-transitorytangible computer-readable medium storing instructions which, whenexecuted by one or more processors, cause a system to perform theabove-disclosed method for design optimization and/or performanceprediction of a material system.

In a further aspect, the invention relates to a computational system fordesign optimization and/or performance prediction of a material system.In one embodiment, the computational system includes one or morecomputing devices comprising one or more processors; and anon-transitory tangible computer-readable medium storing instructionswhich, when executed by the one or more processors, cause the one ormore computing devices to perform the above-disclosed method for designoptimization and/or performance prediction of a material system.

In one aspect, the invention relates to a material system databaseusable for conducting efficient and accurate multiscale modeling of amaterial system, In one embodiment, the material system databaseincludes clusters for a plurality of material systems, each of whichgroups all material points having a same response field within MVE of arespective material system with a unique ID; interaction tensors, eachof which represents interactions of all pairs of the clusters for therespective material system; and response predictions computed based onthe clusters and the interaction tensors.

In one embodiment, the clusters are generated by applying machinelearning to data of response fields of the MVE computed from a materialmodel of the respective material system over a predefined set ofmaterial properties and boundary conditions.

In one embodiment, the interaction tensors are computed with FFT,numerical integration, or FEM

In one embodiment, the responses predictions are obtained by solving agoverning PDE using the clusters and the computed interactions. In oneembodiment, the responses predictions comprise at least effectivestiffness, yield strength, thermal conductivity, damage initiation, andFIP.

In one embodiment, the material system database is configured such thatsome of the response predictions are assigned as a training set fortraining a different machine learning that connects processes/structuresto responses/properties of the material system directly without goingthrough the clustering and interaction computing processes at all; andsome or all of the remaining response predictions are assigned as avalidation set for validating the efficiency and accuracy of themultiscale modeling of the material system.

In one embodiment, the material system database is generated withpredictive reduced order models. In one embodiment, the predictivereduced order models comprise a self-consistent clustering analysis(SCA) model, a virtual clustering analysis (VCA) model, and/or an FEMclustering analysis (FCA) model.

In one embodiment, the material system database is updatable, editable,accessible, and searchable.

In another aspect, the invention relates to a method of applying theabove-disclosed material system database for design optimization and/orperformance prediction of a material system. In one embodiment, themethod includes training a neural network with data of the materialsystem database; and predicting real-time responses during a loadingprocess performed using the trained neueral network, wherein thereal-time responses are used for the design optimization and/orperformance prediction of a material system.

In one embodiment, the method further includes performing a topologyoptimization to design a structure with microstructure information.

In one embodiment, the neural network comprises a feed forward neuralnetwork (FFNN) and/or a convolutional neural network (CNN).

In yet another aspect, the invention relates to a non-transitorytangible computer-readable medium storing instructions which, whenexecuted by one or more processors, cause a system to perform the abovemethod for design optimization and/or performance prediction of amaterial system.

These and other aspects of the invention will become apparent from thefollowing description of the preferred embodiment taken in conjunctionwith the following drawings, although variations and modificationstherein may be affected without departing from the spirit and scope ofthe novel concepts of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The following drawings form part of the present specification and areincluded to further demonstrate certain aspects of the invention. Theinvention may be better understood by reference to one or more of thesedrawings in combination with the detailed description of specificembodiments presented herein. The drawings described below are forillustration purposes only. The drawings are not intended to limit thescope of the present teachings in any way.

FIG. 1 illustrates schematically an overall flow of data through thesystem according to embodiments of the invention. Key: heavy borderedboxes are operations, typically including computer code; these act uponvarious data contained within the parallelograms and typically stored ondisk or in memory depending upon how the operations are embodied. In themore complex operations, information or steps commonly used to produce adesired operation are listed; these appear as numbered, round-corneredboxes. The system starts with a representation of the material system ata finite number of scales. Any particular scale is composed offundamental building blocks, the size of which is defined by acharacteristic length. The composition of the building block is definedby modeling or experimental observations. These are then encoded in adetailed spatial decomposition of the structure (sometimes called a“mesh”), used to describe the location and properties of each phasewithin the building block. Generation of this high-resolutiondescription is termed microstructure generation and may be relevant toany scale. A predefined set (usually carefully selected and simplified)of material properties and boundary conditions are supplied to a directnumerical solver to compute nominal response fields. Unsupervisedmachine learning is applied to these fields to generate clusters thatminimize the distance between points in nominal response space withineach cluster. This produces a reduced representation of the materialbuilding block. The interaction between each cluster (or, the influencethat a unit load applied to one cluster has on other clusters) can becomputed and stored. Using precomputed clusters and interactions, thesolution to the relevant governing partial differential equation (PDE)with arbitrary boundary conditions and material properties (notnecessarily the simplified ones used to compute clusters) is fast in theresponse prediction stage. The results of the response prediction arepassed to the next coarser scale, as the overall response of thatbuilding block, and the process thus proceeds.

FIG. 2 shows discretization of partial differential equations through,panel (a), domain-based decomposition and panel (b), response-baseddecomposition clustering e.g., with unsupervised learning, according toembodiments of the invention.

FIG. 3 shows example material systems with two or more relevant lengthscales that could be modeled by the method and system, according toembodiments of the invention.

FIG. 4 shows an MVE model of a unidirectional carbon fiber reinforcedcomposite containing 36 million voxels, according to embodiments of theinvention. On the left, the blue cylinders represent carbon fiber, andthe red is a polymetric epoxy resin matrix materials. The overall(homogenized) response to an applied load in the y-direction is plottedin the top right, and the local stress contours are plotted on thebottom right. The direct numerical simulation (DNS) process to computethis single load history takes about 200 hours using 80 computerprocessors.

FIG. 5 is an illustration of the 2-scale multiscale problem and itscomplexity using traditional finite element method (top panel), where atthe macroscale, a mesh of a generic part subject to some boundarycondition is shown and at each material point within these elements, amicroscale response corresponding to the behavior of a generic subscale,shown in this example with spherical inclusions, is computed; and anillustration of two-scale multiscale clustering method and itscapability in order reduction (bottom panel), according to embodimentsof the invention. In this illustration, each color represents order 1degrees of freedom, rather than each blue-bounded region within the topillustration.

FIG. 6 shows three-step “training” process, as one possible way tocompute the clusters and interaction tensor for macroscale geometry (toppanel) and microscale geometry (bottom panel), according to embodimentsof the invention.

FIG. 7 shows a three-scale FRP modeling framework, according toembodiments of the invention. The macroscale model is a woven laminatecomposite model, built as a Finite Element mesh. Each integration pointin the macroscale model is represented by a woven MVE, which can havedifferent tow size, tow spacing, and tow angle (the one shown is 90° towangle). Each integration point in the mesoscale model is represented bya UD MVE, which can have different fiber orientation, fiber volumefraction, and matrix-fiber interfacial strength. If all three scales arediscretized with FE meshes, the total DOF for this full-field systemcould be 3.3×10¹⁵. Using the present method and replace mesoscale andmicroscale with ROMs, the total DOF is reduced down to 1.6×10⁶, areduction of nine orders of magnitude.

FIG. 8 shows MVEs generated from a number of sources, according toembodiments of the invention. This example for metallic materials showsgrains measured using x-ray diffraction, reconstructed from astatistical description, and predicted from a processing model. Defects,voids in this case, can be from measurements or predictions, here twoexperimental methods (x-ray tomography and FIB-SEM serial sectioning)are given as examples. These input data are paired (spatially), and usedfor the MVE, resulting in response predictions based directly onexperimental images.

FIG. 9 shows a multiscale cluster-based process according to embodimentsof the invention. Left: the original DNS description of a UD MVE.Center: cluster-based description (showing 2 clusters in the fiberphase, 8 in the matrix), visualized on the underlying voxel mesh. Right:numerical homogenization results for overall stress/strain. The DNSreference solution is shown in red with pointwise 5% error bars. Cf thereduced order solutions with 8 clusters (blue) and 16 clusters (green).

FIG. 10 shows a physics guide NN may be “layered” on top of the MVE ROMaccording to embodiments of the invention: the NN is trained on a largedatabase of rapidly-computed behavior. Once trained, this NN is thoughtto contain microstructural information similar to the ROM, but is muchfaster to evaluate. This alternative makes it practical to conductmicrostructure-based structural optimization and design.

FIG. 11 shows an application example: a composite design, according toembodiments of the invention. The composite structure can be designedwith different microscale structure (e.g., fiber shape) and mesoscalestructure (e.g., fiber orientation in each ply, fiber orientation andfiber shape in each ply, as well as different weave pattern for wovencomposite). Key performance indices (e.g., strength and maximumdeformation under external load) for the composite structure can bepredicted. If those indices do not meet desired criterion, optimizationroutine will be called to update microscale and mesoscale structure inorder to improve performance indices.

FIG. 12 shows woven composite MVE domain decomposition according toembodiments of the invention.

FIG. 13 shows a random grain structure with 35 grains, where each grainis considered a different material phase and grain-by-grain stresspredictions, with progressively more clusters per grain according toembodiments of the invention, as discussed in EXAMPLE 5.

FIG. 14 is an illustration of an implementation with hierarchicalmodeling between mesoscale and microscale, and concurrent modelingbetween macroscale and mesoscale, according to embodiments of theinvention. The UD MVE elastic properties are pre-computed and passed toyarn, constituting to a hierarchical modeling process. When the wovenMVE is under external loading, its response is computed using both yarn(given by the UD MVE) and matrix properties. When the FE model is underexternal loading, its local responses is computed using the woven MVE.The FE model and woven MVE responses are computed in a concurrentfashion, establishing a concurrent modeling scheme. A combinedhierarchical and concurrent modeling is thus implemented.

FIG. 15 shows stress vs. strain curves for six loading directions (wovenis isotropic in the xy plane, thus the yy response is same as the xxresponse, and the xz response is same as the yz response, panel (a)), asdiscussed in EXAMPLE 8; and 3D yield surface visualized with stressstates (magenta asterisks, panel (b)), according to embodiments of theinvention. All asterisks represent a stress state that does not causeyielding of the material. The present method allows one to build yieldsurfaces for various microstructure and generate material constitutiveinformation with minimum effort (around one minute using a personalcomputer). A large woven composite response database can be built toassist design of woven composite against yielding. Given a prioriinformation on maximum service loads, the database will provide allpossible woven microstructure (e.g., yarn geometry and yarn angle) andmaterial constituents (e.g., matrix properties and yarn properties) thatwould prevent yielding to occur.

FIG. 16 shows offline and online stages of two-stage clustering analysismethods according to embodiments of the invention. The offline stagecontains three steps as shown in the figure, which will generate acompressed RVE database. The compressed RVE can be then used to predictmechanical responses of the RVE.

FIG. 17 shows geometry for example problem: a two-phase periodiccomposite represented in 2D with plane strain where blue is a continuousmatrix phase and yellow represents inclusions.

FIG. 18 shows color contours showing clusters distribution in theinclusions (panel a) and the matrix (panel b), according to embodimentsof the invention. Note that clusters need not be spatially connected.

FIG. 19 shows component-wise magnitude plots for a) D_(SCA) ^(IJ), b)D_(VCA) ^(IJ), c) B_(FCA) ^(IJ), according to embodiments of theinvention. Spikes along the diagonal direction for all three interactiontensor surface plots suggest self interaction has more contribution thanthe rest of clusters in cluster-wise stress increment. D_(SCA) ^(IJ) andD_(VCA) ^(IJ) have the similar magnitude along their diagonal directiondue to the homogeneous reference material assumption. B_(FCA) ^(IJ) hasdifferent magnitudes for matrix and inclusion phase along the diagonaldirection, implicitly representing a heterogeneous reference material.

FIG. 20 shows plots for a) D_(SCA) ^(IJ), b) D_(VCA) ^(IJ), c) B_(FCA)^(IJ) in profile, according to embodiments of the invention. Note thatfor FCA the two regions correspond to different physical domains (matrixand inclusion).

FIG. 21 shows plots of σ_(xx) ^(M) vs. ε_(xx) ^(M), according toembodiments of the invention. All three methods performed well, withpredictions laying within 5% of the reference solution.

FIG. 22 shows plots of σ_(yy) ^(M) vs. ε_(xx) ^(M), according toembodiments of the invention. SCA has the best agreement with the DNSsolution in this case, whereas VCA and FCA deviate from the DNSsolution. The causes for such deviation and improvements in VCA and FCAwill be explored in the future.

FIG. 23 shows random samples of strain state, according to embodimentsof the invention; two hundred final states were selected, and fourevenly spaced intermediate steps to reach the final states were recordedfor a total of N_(T)=1,000 samples. All strain states will be applied tothe RVE to generate corresponding stress states.

FIG. 24 is illustration of an FFNN network with one hidden layer for alinear elastic example (panel a), the collective function of the weightsand biases connecting the input layer (green), the hidden layer (blue),and the output layer (red) is that of Young's Modulus E, and astress-strain diagram (panel b), showing how the input strain isinterpreted by the FFNN for a linear elastic case using linearactivation functions and zero biases.

FIG. 25 is illustration of an FFNN with multiple hidden layers; N_(L):index of layers, N_(N)(l): number of neurons in layer l. The formulationof the FFNN is given in Eq. (1-21) with associated interpretation of theFFNN structure. The indices i and j representing neuron id in previouslayer and current layer, e.g., W₁₂ ^(l=2) is the weight between neuron 1in layer l=1 and neuron 2 in layer l=2.

FIG. 26 shows a histogram of the difference between the FFNN σ^(M)predictions and validation data set for all N_(V)=150 samples accordingto embodiments of the invention. Most of the predictions made by FFNNhas an l² norm less than 2×10⁻⁵, showing that the FFNN produce anaccurate prediction of the stress state.

FIG. 27 shows a cross-correlation between SCA and FFNN for x-, y-, andshear-directions according to embodiments of the invention. The solidblack lines are the ground truth shows all perfect predictions shouldlay on those lines. The FFNN-predicted stress components lay around thesolid black lines, and all three cases have correlation coefficients of1, exhibiting very strong predictive confidence.

FIG. 28 shows a stress-strain plot with both the FFNN and SCA solutionsfor the validation data, showing that the FFNN results successfullyreproduced SCA results, according to embodiments of the invention.

FIG. 29 shows an illustration of one dimensional convolutional neuralnetwork with the following setup: padding, convolution, pooling, and afeed forward neural network for regression analysis, according toembodiments of the invention. The first three steps may be repeated.

FIG. 30 shows an illustration of two dimensional convolutional neuralnetwork, according to embodiments of the invention. In the trainingpart, the number of threshold η*=5. After 4 repeats of convolution, thedata will be passed to FFNN. The dimension of input for FFNN becomesN_(fl)=74. The hidden layer in FFNN has 74 neurons. Finally, FFNN givesthree macro strain components.

FIG. 31 shows a histogram of the l² norm for the CNN prediction usingvalidation data points, according to embodiments of the invention. Mostof the predictions made by CNN has an l² norm less than 1×10⁻⁵, showingthe CNN produce an accurate prediction of the strain state. The l² normillustrate the CNN network is able to make a proper prediction of thevalidation data.

FIG. 32 shows a loading prediction by CNN vs. loading applied in SCAusing validation data points, according to embodiments of the invention.The solid black lines are the ground truth shows all perfect predictionsshould lay on those lines. All three cases have correlation coefficientshigher than 0.99, suggesting the trained CNN can provide a good accuracyin predicting applied strain.

FIG. 33 shows a topology optimization setup with FFNN, according toembodiments of the invention. The FFNN will be used to computenon-linear material responses to drive a new design. This replaces theconstitutive law commonly used for the macroscale with a homogenizedresponse of the microstructure for each point in the macroscale.Mathematically, σ^(M) (X^(M))=β(X^(M))

_(FFNN)(ε(X^(M))), ∀X^(M) ∈^(M), as defined in Eq. (1-36).

FIG. 34 shows an optimized beam structures with elastic materialresponses (pane a) and non-linear material responses (panel b),according to embodiments of the invention. Comparing panels a) and b),the joint of all truss members for the non-linear case is locatedtowards the bottom side of the truss structure. This means the materialnon-linear responses plays an important role in determining optimizedtruss structure in this case.

FIG. 35 shows a topology optimization setup with FFNN and CNN, accordingto embodiments of the invention. For each material point within thedesign zone, the FFNN is used to compute the material response, be itlinear or non-linear, considering the effect of microstructure. The CNNis used to incorporate microstructure damage, which will drive theoptimization algorithm for a new design compared to a topologyoptimization with only linear material. Mathematically, this is ∀X^(M)∈^(M) (X^(M)): σ(X)=

_(FFNN) ^(micro)(ε^(M)(X^(M))), ∀X∈(X): d(X^(M))=(

_(CNN) ^(classify)(σ(X))), as defined in Eq. (37). If any d(X) is markedas damaged in the microstructure, the X^(M) point in the design zonecontaining that microstructure will be marked as damaged.

FIG. 36 shows optimized beam structures without damage constraints(panel a) and with damage constraints (panel b), according toembodiments of the invention. In panel (b) there are more truss membersto avoid local stresses concentration that results in damage, asdiscussed in EXAMPLE 1.

FIG. 37 shows ductile materials' microstructures discretized using voxelmeshes with matrix shown in blue and inclusions in red: (panel a)two-dimensional microstructure, (panel b) inside view of athree-dimensional microstructure with a fragmented inclusion surroundedby a debonding void shown in light gray, according to embodiments of theinvention

FIG. 38 shows ductile materials' microstructures, according toembodiments of the invention: (panel a) two-dimensional microstructurediscretized using 8 clusters, (panel b) same two-dimensionalmicrostructure discretized using 65 clusters, and (panel c)three-dimensional microstructure discretized using 217 clusters showingtwo clusters in the matrix phase (two shades of blue), one cluster inthe inclusion phase (red), and one cluster in the void phase (lightgray).

FIG. 39 shows ductile composite with particle volume fraction of 20%(panel a); SCA converges fast to the FFT reference for the overallmechanical response of the RVE under unidirectional tension along the xdirection (panel b); (panel c) a closer look at the dashed rectanglearea in panel (b) shows accurate prediction of SCA with only 16clusters; and (panel d) a CPU time saving of a factor of more than 10³achieved with SCA for the relatively well converged number of clusters,compared with the FFT reference with a 100×100×100 voxel mesh, accordingto embodiments of the invention.

FIG. 40 shows (panel a) SCA is fairly close to the FFT reference for theoverall mechanical response of the porous RVE under uniaxial tensionalong the x direction; (panel b) a closer look at dashed rectangle areain (panel a) showing the relatively slow convergence; (panel c) a CPUtime saving of a factor of more than 10³ is achieved with SCA forrelatively well-converged numbers of clusters, compared with the FFTreference with a 100×100×100 voxel mesh, according to embodiments of theinvention.

FIG. 41 shows clusters (shades of red) within the debonded inclusionshowing (panel a) before fragmentation, and (panel b) afterfragmentation with the missing cluster turned into void, according toembodiments of the invention. Drawing direction is horizontal.

FIG. 42 shows cold drawing and FIP computation results showing theparticle fragments in red, the voids in light gray, according toembodiments of the invention: (panel a) the equivalent plastic strain inthe matrix in undeformed con guration, (panel b) the equivalent plasticstrain in the matrix in deformed con guration after displacementreconstruction, (c) the FIP in the matrix on a new voxel mesh of thedeformed con guration after mesh transfer. Note that the three figuresare not in the same spatial scale, as section height in (panels b and c)is reduced by 45% compared to (panel a).

FIG. 43 shows (panel a) ductile composite with particle volume fractionof 20%; (b) SCA converges fast to the FFT reference for the overallmechanical response of the RVE under unidirectional tension along the xdirection; (panel c) a closer look at the dashed rectangle area in(panel b) shows accurate prediction of SCA with only 16 clusters; (paneld) a CPU time saving of a factor of more than 10³ is achieved with SCAfor the relatively well-converged number of clusters, compared with theFFT reference with a 100×100×100 voxel mesh, according to embodiments ofthe invention.

FIG. 44 illustrates an ICME framework according to embodiments of theinvention.

FIG. 45 shows epoxy matrix according to embodiments of the invention:(panel a) tension stress versus strain curve and (panel b) compressionstress-strain curve.

FIG. 46 shows UD CFRP plaque (approximately 300 mm×300 mm) with fibersoriented in the vertical direction, and the tabbed specimens afterwaterjet cutting, according to embodiments of the invention.

FIG. 47 shows (panel a) field of view with speckle pattern on specimenwith 1=120 mm, w=12.7 mm; and (panel b) region of Interest (coloredarea) for data analysis, according to embodiments of the invention.

FIG. 48 shows UD CFRP hat-section sample according to embodiments of theinvention: (panel a) geometry; (panel b) sample with back plate; and(panel c) dynamic 3 point bending test setup.

FIG. 49 shows UD CFRP coupon FE model setup according to embodiments ofthe invention. In this setup, the material response at each integrationpoint (marked as black cross) is computed using a UD RVE, according toembodiments of the invention.

FIG. 50 shows four candidate microstructure setups in the database forthe UD coupon specimen according to embodiments of the invention. Thedatabase is designed so that users can assign various microstructure tothe macroscale model based on their needs. “els” in this figure standsfor voxel elements in the RVE, according to embodiments of theinvention.

FIG. 51 shows a cross-section of UD CFRP under microscope, withmagnified view shown on the right, according to embodiments of theinvention.

FIG. 52 shows an algorithm flow chart for generating UD RVE using MonteCarlo method, according to embodiments of the invention.

FIG. 53 shows UD RVEs generated by Monte Carlo method, according toembodiments of the invention.

FIG. 54 shows UD RVE clustering process for setup 1 and 2 from FIG. 50,according to embodiments of the invention.

FIG. 55 shows transverse tension results of DNS and ROMs, according toembodiments of the invention. The DNS result (computed from FEM) has anerror bar representing the 95% interval. The results of ROMs computedfrom SCA are within the 95% interval.

FIG. 56 shows UD CFRP coupon off-axial tensile simulation setup,according to embodiments of the invention.

FIG. 57 shows UD CFRP dynamic 3-point bending model setup, according toembodiments of the invention.

FIG. 58 shows UD coupon normal stress vs. normal strain, according toembodiments of the invention. The comparison shows the prediction is ina good match with the test data. The difference between prediction andexperimental result is potentially caused by the microstructurevariation in the real UD material. As stated in EXAMPLE 4, all materiallaws are strictly from experiments, without any calibrated parameters.

FIG. 59 shows contour of(panel a) Y displacement and (panel b) Y strainfield, according to embodiments of the invention. The applieddisplacement on prediction and DIC is 0.9031 mm. In the displacementplots a), two black arrows measure the vertical distance between fringesfrom −0.250 mm and −0.700 mm and the difference is 3.95%. In the grayscale strain contour b), the predicted strain field is comparable to theDIC one. The difference the predictions and the DIC images is caused bymicrostructure variations in the real UD CFRP material, which can causestrain concentration in the real sample.

FIG. 60 shows the coupon crack formation of (panel a) numericalprediction and (panel b) experimental result, according to embodimentsof the invention. Since the numerical model assumes perfect materialswithout microstructure variations, the predicted pattern deviates fromthe test result

FIG. 61 shows a magnified view of coupon local microstructure for markedmacroscale elements at a) d_(y)=1.40 mm; b) d_(y)=1.41 mm; c) d_(y)=1.42mm and local UD RVEs at d) d_(y)=1.40 mm; e) d_(y)=1.41 mm; f)d_(y)=1.42 mm, according to embodiments of the invention. All RVEs areshown in undeformed configuration.

FIG. 62 shows an UD hat-section after the impact from (panel a)prediction (panel b) experiment, according to embodiments of theinvention. The damaged zones on the sidewalls are marked by yellowellipses. It can be observed that the hat-section is being pushedinwards upon the impact, and the impactor will cause a dent on thehat-section.

FIG. 63 shows UD hat-section mesh with magnified view of three columnsof elements in the through-thickness direction, according to embodimentsof the invention. The impactor is hidden for clarity. This setup showsthe location of three columns of elements (12 elements per column,representing all 12 layers of UD laminae) across the thickness directionin the UD hat-section mesh. Each element is represented by one UD RVE,whose fiber orientation matches with the UD layup given. All UD RVEs arecolor-coded according to the fiber orientation.

FIG. 64 shows an isometric view of the hat-section von Mises stresscontour at (panel a) Upon impact; (panel b) d_(z)=4.85 mm; and (panel c)d_(z)=6.85 mm, according to embodiments of the invention. The impactoris hidden for clarity. The present ICME framework is capable ofvisualizing structural level responses and the microstructure responses.In this case, only those elements mentioned in FIG. 63 are visualized.

FIG. 65 shows a magnified view of the hat-section with UD CFRPmicrostructure evolution at (panel a) d_(z)=5.91 mm; (panel b)d_(z)=6.13 mm; and (panel c) d_(z)=6.78 mm, according to embodiments ofthe invention. The UD microstructure damage processes in two markedelements are visualized.

FIG. 66 shows an SCA flow chart for the solving the Lippmann-Schwingerequation, according to embodiments of the invention.

FIG. 67 shows clustering results of the multi-inclusion system based onthe elastic strain concentration tensor A(x), according to embodimentsof the invention. The numbers of clusters in the matrix and inclusionare denoted by k_(m) and k_(i), respectively.

FIG. 68 shows stress-strain curves under uniaxial tension and pure shearloading conditions predicted by the regression-based and projectionbased self-consistent scheme according to embodiments of the invention.The solid lines represent the DNS results for comparison.

FIG. 69 shows schematic of the simple 3D void geometry according toembodiments of the invention.

FIG. 70 shows RVE cut in half to show the (panel a) elastic regionsolution and (panel b) resulting clusters around the inclusion; (panelc) the solution at the onset of plasticity and (panel d) the resultingclusters; and (panel e) after plasticity fully develops, (panel f) theresulting clusters, according to embodiments of the invention.

FIG. 71 shows an overall stress-strain response of the single inclusion,based on different choices of cluster, according to embodiments of theinvention.

FIG. 72 shows RVE including 35 equiaxed, randomly oriented grains (asshown by the inverse pole figure color map) with (a) 20×20×20 and (b)40×40×40 voxel mesh, according to embodiments of the invention.

FIG. 73 shows σ ₃₃ versus ∈ ₃₃ using CPFEM and CPSCA respectively,showing convergence with mesh size and number of clusters, according toembodiments of the invention.

FIG. 74 shows volume plots of S33 for six different cases: (panel a) the203 mesh with CPFEM, (panel b) the 303 mesh with CPFEM, (panel c) the403 mesh with CPFEM clusters, (panel d) the 403 mesh with 35 clusters,(panel e) the 403 with 70 clusters, and (panel f) the 403 mesh with 140clusters, according to embodiments of the invention. The 35 cluster casehas one cluster per grain, whereas the 140 cluster case has fourclusters per grain. Opacity scales with stress level.

FIG. 75 shows Temperature dependence of horizontal shift factor forunfilled and filled rubber where aT, T and T0 stand for horizontal shiftfactor, temperature, and reference temperature respectively according toembodiments of the invention.

FIG. 76 shows measured shear storage modulus of unfilled and filledrubbers according to embodiments of the invention.

FIG. 77 shows measured shear loss modulus of unfilled and filled rubbersaccording to embodiments of the invention.

FIG. 78 shows measured loss tangent of filled rubber according toembodiments of the invention. The peak of the filled rubber is less thanthat of unfilled rubber. The loss tangent in the low frequency region ishigher than that of unfilled rubber.

FIG. shows rubber matrix material hidden according to embodiments of theinvention.

FIG. 80 shows predicted Tan(δ) of filled rubber compared to experimentalresults according to embodiments of the invention.

FIG. 81 shows (panel a) shear storage moduli G′ and (panel b) shear lossmoduli G″ comparison between FFT and experimental results according toembodiments of the invention.

FIG. 82 shows Offline and Online Stages for Filled Rubber according toembodiments of the invention.

FIG. 83 shows clusters of filled rubber in matrix phase and filler phaseaccording to embodiments of the invention. Domain decomposition of2-phase filled rubber into reduced order model represented by 64clusters. Left shows Original voxel mesh for the 2-phase filled rubber;Right shows Compressed 2-phase filled rubber model with 32 clusters inthe matrix phase and 32 clusters in the filler phase.

FIG. 84 shows Tan(δ) of filled rubber computed by SCA according toembodiments of the invention.

FIG. 85 shows (panel a) shear storage moduli G′ plots and (panel b)shear soss moduli G″ plots, according to embodiments of the invention

FIG. 86 shows filled rubber with an interphase of 9.74 nm and clustersof filled rubber in matrix phase, filler phase, and interphase,according to embodiments of the invention. Domain decomposition of3-phase filled rubber into reduced order model represented by 96clusters. Left: Original voxel mesh for the 3-phase filled rubber;Right: Compressed 3-phase filled rubber model with 32 clusters in thematrix phase, 32 clusters in the interphase phase and 32 clusters in thefiller phase.

FIG. 87 shows predicted master curve of filled rubber by FFT and SCA vs.Experimental Results according to embodiments of the invention.

FIG. 88 shows (panel a) Predicted G′ and (panel b) Predicted G″ ofFilled Rubber vs. Experimental Results according to embodiments of theinvention.

FIG. 89 shows Overall diagram of the computational scheme according toembodiments of the invention. Geometry, build process parameters,material, and loading conditions must be specified. These are used toconduct a thermal analysis and a macroscale stress analysis. For eachmaterial point X within these two models, an element-wise sub-model isconstructed to represent a possible state at that point. This uses localthermal history and strain history to determine the microstructure (voidgeometry) and deformation history, respectively. These are used topredict the microscale evolution of state variables such as plasticityand damage, which are homogenized (e.g., by taking the l_(∞) norm of thedomain) and used as element-wise estimators of part-level susceptibilityto failure.

FIG. 90 shows diagram of the defect estimation and database buildingprocess according to embodiments of the invention. In the first part, arelationship between solidification cooling rate (SCR) and void volumefraction (V_(f)) is determined using process modeling and X-ray computedtomography. Second, 2(a) the subsets of the images acquired with X-raytomography are selected on the basis of V_(f), such that the expectedrange of V_(f) for any arbitrary part (with known or predicted thermalhistory) is spanned. 2(b) A database of these possible microstructuresis generated, 2(c) the database include precomputed interaction tensorscomprising the training stage of the reduced order mechanical model.

FIG. 91 shows for each macroscale material point (element, in thiscase), the thermal history and strain history are passed to a microscalesolver; a microstructure is selected from the database based on thethermal history, and deformation boundary conditions are appliedaccording to the strain history, according to embodiments of theinvention. A crystal plasticity based microscale solution is computed,and a homogenized response (e.g., the l_(∞)-norm of the fatigueindicating parameter, if a fatigue problem is chosen) is returned to themacroscale.

FIG. 92 shows (panel a) example component geometry specification (ASTME606 fatigue specimen) (panel b) the two meshes and (panel c) details ofthe specimen meshes, including the difference between stress (left) andthermal (right) meshes, according to embodiments of the invention.

FIG. 93 shows the continuous radius fatigue test specimen, showing thethermal processing predictions partway through the build according toembodiments of the invention.

FIG. 94 shows estimated fatigue lives for multiple realizations of thefatigue coupon, run at different applied strain amplitudes, mimickingexperimental conditions, according to embodiments of the invention. Twodifferent processing conditions (conditions 1 and 2 in Table 6-2) weremodeled.

FIG. 95 shows multiscale features of woven composites from microscale tomacroscale according to embodiments of the invention.

FIG. 96 shows multiscale computational challenge of woven compositestructures three scales; complexity, microstructure according toembodiments of the invention.

FIG. 97 shows SCA scheme for woven composites according to embodimentsof the invention.

FIG. 98 shows concurrent multiscale simulation framework according toembodiments of the invention.

FIG. 99 shows the microstructure of plain weave composites according toembodiments of the invention.

FIG. 100 shows an UD RVE model according to embodiments of the invention

FIG. 101 shows clustering process and results of matrix with 256clusters according to embodiments of the invention. Dimensions are givenin FIG. 99.

FIG. 102 shows clustering process and results of yarns with 64 clustersaccording to embodiments of the invention. For each yarn, clustering isperformed first based on local orientation. The resulting clusters arerefined further using strain concentration A_(m) tensor.

FIG. 103 shows the prediction results given by FEM and SCA (TheSCA-64-128 indicates 64 clusters in matrix and 128 clusters in theyarns) according to embodiments of the invention

FIG. 104 shows the prediction results given by FEM and SCA according toembodiments of the invention.

FIG. 105 shows Hill yield surface calculation workflow according toembodiments of the invention. The 3D yield surfaces are plotted againstthree normal stress components and three shear stress components. Forthe plot against normal stress components, the cross section wheresigma_zz=0 is illustrated. For the plot against shear stress components,the cross section where sigma_xy=0 is plotted.

FIG. 106 shows convergence study for different RVE sizes according toembodiments of the invention.

FIG. 107A shows geometry and mesh of the T-shaped hooking structure:geometry and boundary conditions, according to embodiments of theinvention.

FIG. 107B shows geometry and mesh of the T-shaped hooking structure:mesh model and reduced order model, according to embodiments of theinvention.

FIG. 108 shows Simulation results of the T-shaped hooking structure.

FIG. 109 shows (panel a) An idealized MVE of eight cubic grains withrandomly assigned crystallographic orientation colored according to theinverse pole FIG. 109 in (b); the Euler angles for grain 1 to 8 are(15.2,65.0,184.0), (313.5,28.5,35.7), (102.1,171.5,245.7),(198.3,54,7,340.8), (288.2,1.1,53.1), (300.1,126.2,248.0),(325.7,133.5,39.6), and (47.2,127.1,196.3); (panel b) Inverse pole (IPF)color map showing the orientations in the MVE, with a z-face normal asthe reference direction and three of the Miller indices in a hexagonalclose packed lattice cell; (panel c) Elastic strain component E₁₁calculated using FEM with pure elasticity at 0.02 overall strain underuniaxial tension in the x-direction; (panel d) 128 clusters, 16clusters/grain, obtained from (panel c) using the k-means clusteringmethod, according to embodiments of the invention.

FIG. 110 shows (panel a) macroscale stress-strain curves calculatedusing CPFEM (in black), CPFFT (in red) with different mesh; (panel b) acloser look at the dashed rectangle of (a); (panel c) Macroscalestress-strain curves calculated using CPSCA (in blue) with increasingnumber of clusters per grain where the high-fidelity CPFEM and CPFFTresults are shown as reference; and (panel d) a closer look at thedashed rectangle of (panel c), according to embodiments of theinvention. The green shading indicates an area within 5% of thereference solution and the red within 1% of the reference solution.

FIG. 111 shows (panel a) The errors of 0.2% offset and 0.4% offsetstress as a function of number clusters used per grain; (panel b) CPUtime versus number of clusters (or voxels) per grain for CPSCA comparedwith using CPFEM and CPFFT, according to embodiments of the invention.Note that all simulations were run on Intel Haswell E5-2680v32.5 GHzprocessors.

FIG. 112 shows multiple realizations of thereconstruction-computation-evaluation loop generates amicrostructure-property database according to embodiments of theinvention.

FIG. 113 shows example MVEs with about 90 grains and corresponding grainsize histograms according to embodiments of the invention. The grainsare sampled with (panel a) no texture, (panel b) (0,0,0) preferred,(panel c) (90,0,0) preferred, and (panel d) (90,90,0) preferred, all ofwhich are colored by the inverse pole of z-direction.

FIG. 114 shows predicted stress-strain curves of 50 MVEs each sampledwith (panel a) no texture, (panel b) (0,0,0) preferred, (panel c)(90,45,0) preferred, and (panel d) (90,90,0) preferred, according toembodiments of the invention.

FIG. 115 shows distribution plots of predicted effective Young's modulus(panel a) and 0.2% offset effective yield strength (panel b) ofdifferent texture cases. Red lines show mean and one standard deviationspread, according to embodiments of the invention.

FIG. 116 shows example MVEs with increasing average grain size shows(panel a) ESD=13.2 μm, (panel b) ESD=19.7 μm, (panel c) ESD=26.6 μm,(panel d) ESD=35.9 μm, according to embodiments of the invention. Thegrains are colored by IPF.

FIG. 117 shows distribution plots of predicted effective Young's modulus(panel a) and 0.2% offset yield strength (panel b) with differentaveraged grain size, according to embodiments of the invention.

FIG. 118 shows a schematic of (panel a) the rolling process and (panelb) its concurrent multiscale simulation method, according to embodimentsof the invention. Given deformation gradient F⁰ for each material pointin the part, the first Piola-Kirchoff stress P⁰ is obtained by solving apolycrystalline MVE problem.

FIG. 119 shows contour of shear stress component σ₁₂ (unit: MPa) atrolling time=0.08 seconds plotted on the deformed configuration,simulated with 3D elements (a and b) and plane strain elements (c),according to embodiments of the invention. 3D simulation predicts lowerdeformation in the rolling direction and higher extreme shear stress.

FIG. 120 shows history of σ₁₂ for each integration point of the threeelements indicated in panel (panel c) of FIG. 119, according toembodiments of the invention. The shear stress value of elements closerto the contact region tend to alternate more times and with higheramplitude.

FIG. 121 shows snapshots of macroscale shear stress contour, andmicroscale equivalent plastic strain and (0001) pole figures associatedwith the three elements, according to embodiments of the invention. Awayfrom the contact surface the associated MVE experiences less rotationand shear, and more compression and plastic strain.

FIG. 122 shows uniaxially tensile (panel a) and compressive (panel b)stress-strain curves for Epon 825 deformed at a strain rate of 5×10⁸ s⁻¹for different temperatures, according to embodiments of the invention.

FIG. 123 shows yield surfaces obtained for different temperaturesaccording to embodiments of the invention, where the points aresimulation data and the lines are theoretical prediction using Eq.(9-1).

FIG. 124 shows effect of crosslink degree and component ratio on thestress-strain behavior by using epoxy 3501-6 as a model system,according to embodiments of the invention.

FIG. 125 shows interphase property characterization, according toembodiments of the invention. (panel a) Schematic of the cross-sectionincluding the interphase region (yellow). (panel b) Variation of Young'smodulus or strength inside the interphase region.

FIG. 126 shows Prepreg's deformation mechanisms during preforming,according to embodiments of the invention.

FIG. 127 shows the experiment setting for the uniaxial tension andbias-extension tests, according to embodiments of the invention.

FIG. 128 shows engineering strain-stress curves from the uniaxialtension tests, according to embodiments of the invention.

FIG. 129 shows bias-extension test results for different specimen sizes:(panel a) original load-displacement curves and (panel b) normalizedload-displacement curves, according to embodiments of the invention.

FIG. 130 shows bias-extension test results for (panel a) differenttemperatures and (panel b) different tensile rates, according toembodiments of the invention.

FIG. 131 shows validation for the kinematic assumption of thebias-extension tests from (panel a) Green strain field obtained from DICand (panel b) average Green strain comparison in the central region.

FIG. 132 shows (panel a) schematic of the bending test setup and (panelb) the shape of prepreg at 50° C., according to embodiments of theinvention.

FIG. 133 shows experimental setup for the prepreg-prepreg interactiontest apparatus, according to embodiments of the invention.

FIG. 134 shows schematic of the interaction measuring experimentalapparatus, according to embodiments of the invention.

FIG. 135 shows force and interaction factor results from the test underthe conditions of 70° C., 5 mm/s, and 0/90/0/90 fiber orientation,according to embodiments of the invention.

FIG. 136 shows steady-state interaction factor in a periodical variationsubjected to the test conditions of 50° C., 15 mm/s, for the 0/90/0/90fiber orientation, according to embodiments of the invention.

FIG. 137 shows interaction and stick-slip strength at varioustemperatures subjected to different (panel a) relative motion speeds and(panel b) fiber orientations, according to embodiments of the invention.

FIG. 138 is an illustration of the prepreg structure via (panel a) realproduct photo and (panel b) model generated by the software TexGen,according to embodiments of the invention.

FIG. 139 shows seometry and forces of the simulated two 2×2 twill fabricinterface, according to embodiments of the invention.

FIG. 140 shows (panel a) experimental and numerical interaction factorcomparison at various temperatures and 10 mm/s; and (panel b) a zoom-into 60° c. and 70° c. for clear illustration, according to embodiments ofthe invention.

FIG. 141 shows experimental and numerical interaction factor comparisonat various speeds and 60° C. temperature, according to embodiments ofthe invention. The points are moved away with the input speedsartificially to better differentiate between the data.

FIG. 142 shows fast Fourier transformation (FFT) results of thenumerical and experimental data, according to embodiments of theinvention.

FIG. 143 shows stress analysis in the non-orthogonal material model,according to embodiments of the invention.

FIG. 144 shows calculation flowchart of the LS-DYNA MAT 293, accordingto embodiments of the invention.

FIG. 145 shows TexGen rough geometry model with the thickness of theprepreg as 1.2 mm, according to embodiments of the invention: (panel a)the structure, and (panel b) the cross-section of the correspondingmesh.

FIG. 146 shows prepreg RVE compression in FE software Abaqus: two rigidplates are introduced to adjust the RVE thickness, according toembodiments of the invention.

FIG. 147 is an illustration of the yarn cross-section deformation uponcompression along the width direction: (panel a) real materialdeformation mode, (panel b) FE deformation mode with transverselyisotropic material model, and (panel c) FE deformation mode withdecoupled material model in FE, according to embodiments of theinvention.

FIG. 148 shows one bias-extension RVE simulation example: (panel a)illustration of the von Mises stress contour on the RVE; (panel b)comparison of the simulation and experimental true shear stress,according to embodiments of the invention.

FIG. 149 shows modular Bayesian calibration, according to embodiments ofthe invention: The approach has four stages and enables estimating thepotential simulator bias as well as the joint posterior distribution ofthe calibration parameters.

FIG. 150 shows marginal posterior distributions on the calibrationparameters, according to embodiments of the invention: The posterior andprior are indicated with solid blue and dotted red lines, respectively.

FIG. 151 shows posterior mean of the responses, according to embodimentsof the invention: (panel a) normal stress as a function of normal truestrain along the yarns for two different shear angles; (panel b) shearstress as a function of normal true strain along the yarns for twodifferent fabric shear angles; (panel c) uniaxial tension test used incalibration vs. our predictions; and (panel d) bias extension test whichis not used in calibration vs. our predictions.

FIG. 152 shows flowchart of the developed multiscale preformingsimulation method, according to embodiments of the invention: TheBayesian calibration utilizes the RVE and experiments to obtain the yarnproperties and the mesoscale stress emulator. The stress emulator isthen implemented into the non-orthogonal material model for macroscopicpreformation simulation.

FIG. 153 shows schematic of different scales in CFRP, according toembodiments of the invention.

FIG. 154 shows microscopy image of unidirectional UD cross-section,according to embodiments of the invention.

FIG. 155 shows UD RVE isometric view with fiber in black and matrix ingrey, according to embodiments of the invention.

FIG. 156 shows woven RVE geometry, according to embodiments of theinvention.

FIG. 157 shows an illustration of yarn angle, yarn fiber volumefraction, and fiber misalignment in yarn, according to embodiments ofthe invention.

FIG. 158 shows effect of yarn angle on components of woven stiffnessmatrix, according to embodiments of the invention.

FIG. 159 shows Illustration of clusters in the fiber phase and thematrix phase, according to embodiments of the invention.

FIG. 160 shows schematic of UD concurrent multiscale modeling, accordingto embodiments of the invention.

FIG. 161 shows (panel a) UD coupon modeling geometry and (panel b)boundary condition, according to embodiments of the invention.

FIG. 162 shows Von Mises stress contour of the coupon model and localRVEs a) before crack initiation b) after crack formation, according toembodiments of the invention.

FIG. 163 shows (panel a) UD hat-section crash model setup; (panel b)impactor force vs. time plot, according to embodiments of the invention.

FIG. 164 shows schematic of concurrent simulation of UD dynamic 3-ptbending, according to embodiments of the invention.

FIG. 165 shows 3-scale concurrent multiscale modeling setup, accordingto embodiments of the invention.

FIG. 166 shows geometry of woven RVE in concurrent multiscale modeling,according to embodiments of the invention.

FIG. 167 shows engineering shear stress vs. engineering shear strain of3-scale and 2-scale single element simple shear, according toembodiments of the invention.

FIG. 168 shows woven bias tension test setup, according to embodimentsof the invention.

FIG. 169 shows (panel a) woven bias sample σ₂₂ contour with 0.4 mmapplied displacement to the top tab; (panel b) σ₂₂ vs ε₂₂ curves ofconcurrent biaxial tension and three sets of test data, according toembodiments of the invention.

FIG. 170 shows multiscale structure schematic view of a four-scale wovenfiber composite with polymer matrix, according to embodiments of theinvention. In computational modeling of this structure, each integrationpoint at any scale is a realization of a structure at the next finerscale.

FIG. 171 shows demonstration of our approach for s two-scale structure,according to embodiments of the invention: spatial random processes(SRPs) are employed for generating spatial variations that are connectedthrough the top-down sampling procedure.

FIG. 172 shows the macroscopic cured woven laminate structure studied inour work, according to embodiments of the invention. (panel a) thedeformed structure, where the light blue lines indicate the fiberorientation, and the dimensions are scaled for a clearer representation;(panel b) the deterministic spatial variations of yarn angle obtainedfrom simulating a perfectly manufactured composite; (panel c) Von Misesstress field corresponding to Case 9; (panel d) the random spatialvariations of yarn angle corresponding to one of the realizations ofCase 1; and (panel e) the random spatial variations of θ ¹ correspondingto one of the realizations of Case 3.

FIG. 173 shows fiber misalignment angles, according to embodiments ofthe invention. The zenith and azimuth angles characterize the fibermisalignment angle with respect to the local orthogonal frame on theyarn cross-section.

FIG. 174 shows coupling the uncertainty sources across the scales,according to embodiments of the invention: The spatial variations of vand θ at the macroscale are connected to those at the finer scales. Forbrevity, the coupling is illustrated only for the average values for thetwo quantities (i.e., the mean of the RFs: β=[β_(v), β_(θ)]=[v ², θ ²]).

FIG. 175 shows effect of average values on the effective moduli of awoven RVE, according to embodiments of the invention: (panel a) Effectof fiber volume fraction and, (panel b) Effect of misalignment. Eachpoint on these figures indicates the average value over 20 simulations.The Case IDs in (panel b) are defined in Table 9-17. The referencesolution refers to a case where there is no misalignment.

FIG. 176 shows prediction error as a function of the number of trainingsamples, according to embodiments of the invention: As the number oftraining samples increases, the accuracy of the MRGP metamodel inpredicting the elements of the stiffness matrix of the mesoscale wovenRVE increases.

FIG. 177 shows screenshots of graphical user interfaces, according toembodiments of the invention: (panel a) optimal Latin hypercube samplinguser interface. (panel b) Gaussian process modeling user interface.

FIG. 178 shows global and local response of the macrostructure to thespatial variations, according to embodiments of the invention: (panel a)reaction force, (panel b) mean stress at the mid-section, and (panel c)standard deviation of the stress field at the mid-section. for cases 1through 8 in panels (b) and (c), the curves are based on 20 independentsimulations.

FIG. 179 shows (panel a) a typical fiber volume fraction (VF) contour,yellow color represents high VF (about 0.6), and blue color representslow VF (about 0.4); and (panel b) the use of the location of splittinglines to represent the non-stationarity, according to embodiments of theinvention.

FIG. 180 shows (panel a) probability density estimate of the inter-linedistances of 3 of the samples; and (panel b) Probability densityestimates of 3 batches of the samples (each batch contains 10 samples),according to embodiments of the invention.

FIG. 181 shows two examples of reconstructions, according to embodimentsof the invention.

FIG. 182 shows the segmented linear regression process, according toembodiments of the invention.

FIG. 183 shows an illustration of the local waviness found by theproposed method, according to embodiments of the invention.

FIG. 184 shows fiber waviness characterization of a large image,according to embodiments of the invention.

FIG. 185 shows some reconstructions and their correspondingperiodograms, according to embodiments of the invention.

FIG. 186 shows geometry of the double-dome punch and the binder,according to embodiments of the invention.

FIG. 187 shows double-dome preforming test setup, according toembodiments of the invention: (panel a) the press for the preforming,and (panel b) the prepreg temperature history plot. The plot indicatesthat the prepreg temperature drops rapidly from the initial 70° C. toaround 23° C. when it is placed under the press.

FIG. 188 shows double-dome preforming simulation setup, according toembodiments of the invention.

FIG. 189 shows simulation and experimental results comparison ofdeformed geometry and yarn angle distribution for double-dome preformedpart of ±45° single layer woven prepreg.

FIG. 190 shows preforming simulation, according to embodiments of theinvention: (panel a) final part shapes and yarn angle distributions, and(panel b) punch force comparison. In panel (a), A-E points indicate yarnangle measuring positions.

FIG. 191 shows double-dome comparison with different initial prepreglayer orientations, according to embodiments of the invention: (panel a)0/90, (panel b) −45/+45, and (panel c) 0/90/−45/+45. The simulationresults are shown in the top half while the experimental ones are in thebottom half. The silver lines on the experimental results indicate thedirections of the warp and weft yarns.

FIG. 192 shows double-dome warp-weft yarn angle distribution results,according to embodiments of the invention: (panel a) sampling points onthe 0/90 prepreg layer, (panel b) sampling points on the −45/+45sampling points, (panel c) angle comparison for the single layer 0/90fiber orientation preforming, (panel d) angle comparison for the singlelayer −45/+45 fiber orientation preforming, (panel e) angle comparisonfor the 0/90 fiber orientation layer in the 0/90/−45/+45 double layerpreforming, and (panel f) angle comparison for the −45/+45 fiberorientation layer in the 0/90/−45/+45 double layer preforming.

FIG. 193 shows folding of the prepreg after low temperature preformingespecially at the edge, according to embodiments of the invention.

DETAILED DESCRIPTION OF THE INVENTION

The present invention will now be described more fully hereinafter withreference to the accompanying drawings, in which exemplary embodimentsof the present invention are shown. The present invention may, however,be embodied in many different forms and should not be construed aslimited to the embodiments set forth herein. Rather, these embodimentsare provided so that this disclosure will be thorough and complete, andwill fully convey the scope of the invention to those skilled in theart. Like reference numerals refer to like elements throughout.

The terms used in this specification generally have their ordinarymeanings in the art, within the context of the invention, and in thespecific context where each term is used. Certain terms that are used todescribe the invention are discussed below, or elsewhere in thespecification, to provide additional guidance to the practitionerregarding the description of the invention. For convenience, certainterms may be highlighted, for example using italics and/or quotationmarks. The use of highlighting and/or capital letters has no influenceon the scope and meaning of a term; the scope and meaning of a term arethe same, in the same context, whether or not it is highlighted and/orin capital letters. It will be appreciated that the same thing can besaid in more than one way. Consequently, alternative language andsynonyms may be used for any one or more of the terms discussed herein,nor is any special significance to be placed upon whether or not a termis elaborated or discussed herein. Synonyms for certain terms areprovided. A recital of one or more synonyms does not exclude the use ofother synonyms. The use of examples anywhere in this specification,including examples of any terms discussed herein, is illustrative onlyand in no way limits the scope and meaning of the invention or of anyexemplified term. Likewise, the invention is not limited to variousembodiments given in this specification.

It will be understood that, although the terms first, second, third,etc. may be used herein to describe various elements, components,regions, layers and/or sections, these elements, components, regions,layers and/or sections should not be limited by these terms. These termsare only used to distinguish one element, component, region, layer orsection from another element, component, region, layer or section. Thus,a first element, component, region, layer or section discussed below canbe termed a second element, component, region, layer or section withoutdeparting from the teachings of the present invention.

It will be understood that, as used in the description herein andthroughout the claims that follow, the meaning of “a”, “an”, and “the”includes plural reference unless the context clearly dictates otherwise.Also, it will be understood that when an element is referred to as being“on,” “attached” to, “connected” to, “coupled” with, “contacting,” etc.,another element, it can be directly on, attached to, connected to,coupled with or contacting the other element or intervening elements mayalso be present. In contrast, when an element is referred to as being,for example, “directly on,” “directly attached” to, “directly connected”to, “directly coupled” with or “directly contacting” another element,there are no intervening elements present. It will also be appreciatedby those of skill in the art that references to a structure or featurethat is disposed “adjacent” to another feature may have portions thatoverlap or underlie the adjacent feature.

It will be further understood that the terms “comprises” and/or“comprising,” or “includes” and/or “including” or “has” and/or “having”when used in this specification specify the presence of stated features,regions, integers, steps, operations, elements, and/or components, butdo not preclude the presence or addition of one or more other features,regions, integers, steps, operations, elements, components, and/orgroups thereof.

Furthermore, relative terms, such as “lower” or “bottom” and “upper” or“top,” may be used herein to describe one element's relationship toanother element as illustrated in the figures. It will be understoodthat relative terms are intended to encompass different orientations ofthe device in addition to the orientation shown in the figures. Forexample, if the device in one of the figures is turned over, elementsdescribed as being on the “lower” side of other elements would then beoriented on the “upper” sides of the other elements. The exemplary term“lower” can, therefore, encompass both an orientation of lower andupper, depending on the particular orientation of the figure. Similarly,if the device in one of the figures is turned over, elements describedas “below” or “beneath” other elements would then be oriented “above”the other elements. The exemplary terms “below” or “beneath” can,therefore, encompass both an orientation of above and below.

Unless otherwise defined, all terms (including technical and scientificterms) used herein have the same meaning as commonly understood by oneof ordinary skill in the art to which the present invention belongs. Itwill be further understood that terms, such as those defined in commonlyused dictionaries, should be interpreted as having a meaning that isconsistent with their meaning in the context of the relevant art and thepresent disclosure, and will not be interpreted in an idealized oroverly formal sense unless expressly so defined herein.

As used in this disclosure, “around”, “about”, “approximately” or“substantially” shall generally mean within 20 percent, preferablywithin 10 percent, and more preferably within 5 percent of a given valueor range. Numerical quantities given herein are approximate, meaningthat the term “around”, “about”, “approximately” or “substantially” canbe inferred if not expressly stated.

As used in this disclosure, the phrase “at least one of A, B, and C”should be construed to mean a logical (A or B or C), using anon-exclusive logical OR. As used herein, the term “and/or” includes anyand all combinations of one or more of the associated listed items.

The methods and systems will be described in the following detaileddescription and illustrated in the accompanying drawings by variousblocks, components, circuits, processes, algorithms, etc. (collectivelyreferred as “members”). These members may be implemented usingelectronic hardware, computer software, or any combination thereof.Whether such elements are implemented as hardware or software dependsupon the particular application and design constraints imposed on theoverall system. By way of example, a member, or any portion of anmember, or any combination of members may be implemented as a“processing system” that includes one or more processors. Examples ofprocessors include microprocessors, microcontrollers, graphicsprocessing units (GPUs), central processing units (CPUs), applicationprocessors, digital signal processors (DSPs), reduced instruction setcomputing (RISC) processors, systems on a chip (SoC), basebandprocessors, field programmable gate arrays (FPGAs), programmable logicdevices (PLDs), state machines, gated logic, discrete hardware circuits,and other suitable hardware configured to perform the variousfunctionality described throughout this disclosure. One or moreprocessors in the processing system may execute software. Software shallbe construed broadly to mean instructions, instruction sets, code, codesegments, program code, programs, subprograms, software components,applications, software applications, software packages, routines,subroutines, objects, executables, threads of execution, procedures,functions, etc., whether referred to as software, firmware, middleware,microcode, hardware description language, or otherwise.

Accordingly, in one or more example embodiments, the functions describedmay be implemented in hardware, software, or any combination thereof. Ifimplemented in software, the functions may be stored on or encoded asone or more instructions or code on a computer-readable medium.Computer-readable media includes computer storage media. Storage mediamay be any available media that can be accessed by a computer. By way ofexample, and not limitation, such computer-readable media can comprise arandom-access memory (RAM), a read-only memory, an electrically erasableprogrammable read-only memory (EEPROM), optical disk storage, magneticdisk storage, other magnetic storage devices, combinations of theaforementioned types of computer-readable media, or any other mediumthat can be used to store computer executable code in the form ofinstructions or data structures that can be accessed by a computer.

The description below is merely illustrative in nature and is in no wayintended to limit the invention, its application, or uses. The broadteachings of the invention can be implemented in a variety of forms.Therefore, while this invention includes particular examples, the truescope of the invention should not be so limited since othermodifications will become apparent upon a study of the drawings, thespecification, and the following claims. For purposes of clarity, thesame reference numbers will be used in the drawings to identify similarelements. It should be understood that one or more steps within a methodmay be executed in different order (or concurrently) without alteringthe principles of the invention.

Microstructured materials are aggregates of individual components. Inmulti-component materials, the components, which are considered asmaterial building blocks, aggregate or self-assemble to form a complexstructures or conformations at multiple scales. Multiscale modelingmethods attempt to faithfully capture the emergent complex behaviors onseveral length- and time-scales.

One of the objectives of this invention is to provide data-drivenrepresentation and clustering discretization methods and systems, whichis a data-driven domain decomposition approach that is suitable toaccelerate the numerical simulation of the microstructure responses. Theconcept here is similar to the Lebesgue Integral, illustrated in FIG. 2,which depends on grouping nearby response (y-axis) rather than nearbydomain (x-axis) used for the Riemann Integral. This decompositionapproach significantly reduces overall computational complexity of aproblem, especially in multiscale analysis.

In one aspect of the invention, a method for design optimization and/orperformance prediction of a material system includes generating arepresentation of the material system at a number of scales, wherein therepresentation at a scale comprises microstructure volume elements (MVE)that are building blocks of the material system at said scale;collecting data of response fields of the MVE computed from a materialmodel of the material system over a predefined set of materialproperties and boundary conditions; applying machine learning to thecollected data of response fields to generate clusters that minimize adistance between points in a nominal response space within each cluster;computing an interaction tensor of interactions of each cluster witheach of the other clusters; and solving an governing partialdifferential equation (PDE) using the generated clusters and thecomputed interactions to result in a response prediction that is usablefor the design optimization and/or performance prediction of thematerial system.

In one embodiment, the method further comprises passing the resultingresponse prediction to a next coarser scale as an overall response ofthat building block, and iterating the process until a final scale isreached.

In one embodiment, the building blocks are defined by materialproperties and structural descriptors obtained by modeling orexperimental observations and encoded in a domain decomposition ofstructures for identifying locations and properties of each phase withinthe building blocks.

In one embodiment, the structural descriptors comprise characteristiclength and geometry.

In one embodiment, the boundary conditions are chosen to satisfy theHill-Mandel condition.

In one embodiment, the collected data of response fields comprise astrain concentration tensor, a deformation concentration tensor, stresstensor (e.g., PK-I stress, Cauchy stress), plastic strain tensor,thermal gradient, or the like.

In one embodiment, the machine learning comprises unsupervised machinelearning and/or supervised machine learning.

In one embodiment, the machine learning is performed with aself-organizing mapping (SOM) method, a k-means clustering method, orthe like.

In one embodiment, the clusters are generated by marking all materialpoints that have the same response field within the representation ofthe material system with a unique ID and grouping material points withthe same ID.

In one embodiment, the generated clusters are a reduced representationof the material system, which reduces the number of degrees of freedomrequired to represent the material system.

In one embodiment, the generated clusters are a reduced order MVE of thematerial system.

In one embodiment, the computed interaction tensor is for all pairs ofthe clusters.

In one embodiment, said computing of the interaction tensor is performedwith fast Fourier transform (FFT), numerical integration, or finiteelement method (FEM).

In one embodiment, the PDE is a Lippmann-Schwinger equation. In oneembodiment, said solving of the PDE is performed with arbitrary boundaryconditions and/or material properties.

In one embodiment, the collected data of response fields, the generatedclusters, and/or the computed interaction tensor are saved in one ormore material system databases.

In one embodiment, said solving of the PDE is performed in real-time byaccessing the one or more material system databases for the generatedclusters and the computed interactions.

In another aspect of the invention, a method for design optimizationand/or performance prediction of a material system includes performingan offline data compression, wherein original microstructure volumeelements (MVE) of building blocks of the material system are compressedinto clusters, and an interaction tensor of interactions of each clusterwith each of the other clusters is computed; and solving an governingPDE using the clusters and the computed interactions to result in aresponse prediction that is usable for the design optimization and/orperformance prediction of the material system.

In one embodiment, the method further includes passing the resultingresponse prediction to a next coarser scale as an overall response ofthat building block, and iterating the process until a final scale isreached.

In one embodiment, the building blocks are defined by materialproperties and structural descriptors obtained by modeling orexperimental observations and encoded in a domain decomposition ofstructures for identifying locations and properties of each phase withinthe building blocks.

In one embodiment, the structural descriptors comprise characteristiclength and geometry.

In one embodiment, when more than one scale is involved with the reducedorder response prediction, the method is named MultiresolutionClustering Analysis (MCA).

In one embodiment, the boundary conditions are chosen to satisfy theHill-Mandel condition.

In one embodiment, said performing the offline data compressioncomprises collecting data of response fields of the MVE computed from amaterial model of the material system over a predefined set of materialproperties and boundary conditions; applying machine learning to thecollected data of response fields to generate clusters that minimize adistance between points in a nominal response space within each cluster;and computing the interaction tensor is for all pairs of the clusters.

In one embodiment, the collected data of response fields comprise astrain concentration tensor, a deformation concentration tensor, stresstensor (e.g., PK-I stress, Cauchy stress), plastic strain tensor,thermal gradient, or the like.

In one embodiment, the machine learning comprises unsupervised machinelearning and/or supervised machine learning.

In one embodiment, the machine learning is performed with an SOM method,a k-means clustering method, or the like.

In one embodiment, the clusters are generated by marking all materialpoints having the same response field within the representation of thematerial system with a unique ID and grouping material points with thesame ID.

In one embodiment, the clusters are a reduced representation of thematerial system, which reduces the number of degrees of freedom requiredto represent the material system.

In one embodiment, the clusters are a reduced order MVE of the materialsystem.

In one embodiment, said computing the interaction tensor is performedwith FFT, numerical integration, or FEM.

In one embodiment, the PDE is a Lippmann-Schwinger equation. h oneembodiment, said solving the PDE is performed with arbitrary boundaryconditions and material properties.

In one embodiment, the collected data of response fields, the generatedclusters, and/or the computed interaction tensor are saved in one ormore material system databases.

In one embodiment, said solving the PDE is performed with onlineaccessing the one or more material system databases for the generatedclusters and the computed interactions.

In yet another aspect, the invention relates to a material systemdatabase usable for conducting efficient and accurate multiscalemodeling of a material system, In one embodiment, the material systemdatabase includes clusters for a plurality of material systems, each ofwhich groups all material points having a same response field within MVEof a respective material system with a unique ID; interaction tensors,each of which represents interactions of all pairs of the clusters forthe respective material system; and response predictions computed basedon the clusters and the interaction tensors.

In one embodiment, the clusters are generated by applying machinelearning to data of response fields of the MVE computed from a materialmodel of the respective material system over a predefined set ofmaterial properties and boundary conditions.

In one embodiment, the interaction tensors are computed with FFT,numerical integration, or FEM.

In one embodiment, the responses predictions are obtained by solving agoverning PDE using the clusters and the computed interactions. In oneembodiment, the responses predictions comprise at least effectivestiffness, yield strength, thermal conductivity, damage initiation, andFIP.

In one embodiment, the material system database is configured such thatsome of the response predictions are assigned as a training set fortraining a different machine learning that connects processes/structuresto responses/properties of the material system directly without goingthrough the clustering and interaction computing processes at all; andsome or all of the remaining response predictions are assigned as avalidation set for validating the efficiency and accuracy of themultiscale modeling of the material system.

In one embodiment, the material system database is generated withpredictive reduced order models. In one embodiment, the predictivereduced order models comprise a self-consistent clustering analysis(SCA) model, a virtual clustering analysis (VCA) model, and/or an FEMclustering analysis (FCA) model.

In one embodiment, the material system database is updatable, editable,accessible, and searchable.

In a further aspect, the invention relates to a method of applying theabove-disclosed material system database for design optimization and/orperformance prediction of a material system. In one embodiment, themethod includes training a neural network with data of the materialsystem database; and predicting real-time responses during a loadingprocess performed using the trained neueral network, wherein thereal-time responses are used for the design optimization and/orperformance prediction of a material system.

In one embodiment, the method further includes performing a topologyoptimization to design a structure with microstructure information.

In one embodiment, the neural network comprises a feed forward neuralnetwork (FFNN) and/or a convolutional neural network (CNN).

In one aspect, the invention relates to a non-transitory tangiblecomputer-readable medium storing instructions which, when executed byone or more processors, cause a system to perform the above-disclosedmethods for design optimization and/or performance prediction of amaterial system.

In another aspect, the invention relates to a computational system fordesign optimization and/or performance prediction of a material system.In one embodiment, the computational system includes one or morecomputing devices comprising one or more processors; and anon-transitory tangible computer-readable medium storing instructionswhich, when executed by the one or more processors, cause the one ormore computing devices to perform the above-disclosed method for designoptimization and/or performance prediction of a material system.

The advantages and specific applications of the invented methods andsystems are briefed as follows, while the details of them are discussedin EXAMPLES 1-9 following the section.

A methodology for constructing microstructure material databases forfast microstructure-derived response prediction has been developed. Themethod takes materials data and compresses it using unsupervised machinelearning to create a “clustering discretization” in effect a speciallydesigned database that is suitable to conduct efficient and accuratemultiscale modeling of arbitrary material systems. Because the methoddescribed here relies on capturing and combining the fundamentalbuilding blocks of the material and its response, any complex and/orhierarchical material systems can be accurately and efficiently modeled,thus, the method is material agnostic. The method described here canalso be applicable to prediction of many physical phenomena that sharesimilar underlying mathematical descriptions. While the examplesprovided are related to the material behavior of solids subjected tomechanical loads, this should not be thought of as a fundamentalrestriction; the method could well be applied to predict a range ofeffective physics, including, but not limited to, electric and magneticproperties. In effect, the method might be used to solve arbitrarycomputational homogenization problems, or problems involving thesolution of partial differential equations.

FIG. 1 illustrates schematically an overall flow of data through thesystem according to embodiments of the invention, where heavy borderedboxes, e.g., 110, 120, 130, 140, 150, are operations, typicallyincluding computer codes, and these operations act upon various datacontained within the parallelograms, e.g., 115, 125, 135, 145, 155, andtypically stored on disk or in memory depending upon how the operationsare embodied. In the more complex operations, information or stepscommonly used to produce a desired operation are listed; these appear asnumbered, round-cornered boxes. The system starts with a representationof the material system at a finite number of scales. Any particularscale is composed of fundamental building blocks, the size of which isdefined by a characteristic length. The composition of the buildingblocks is defined by modeling or experimental observations. These arethen encoded in a detailed spatial decomposition of the structure(sometimes called a “mesh”), used to describe the location andproperties of each phase within the building blocks. Generation of thishigh-resolution description is termed as microstructure generation andmay be relevant to any scale. A predefined set (usually carefullyselected and simplified) of material properties and boundary conditionsare supplied to a direct numerical solver to compute nominal responsefields. Unsupervised machine learning is applied to these fields togenerate clusters that minimize the distance between points in nominalresponse space within each cluster. This produces a reducedrepresentation of the material building block. The interaction betweeneach cluster (or, the influence that a unit load applied to one clusterhas on other clusters) can be computed and stored. Using precomputedclusters and interactions, the solution to the relevant governingpartial differential equation (PDE) with arbitrary boundary conditionsand material properties (not necessarily the simplified ones used tocompute clusters) is fast in the response prediction stage. The resultsof the response prediction are passed to the next coarser scale, as theoverall response of that building block, and the process thus proceeds.

A multiscale analysis is envisioned to include models that capturematerial behavior at each relevant length scale passing information tothe next higher length scale—until the final scale is reached. Thisappears on the far left of FIG. 1

The next step shown in FIG. 1 involves generation of an MVE, or materialrepresentation. A reasonable MVE, or set of MVEs depending on the numberof scales involved, for any given problem might be derived from manydifferent sources. This method can be used with MVEs derived fromprocess simulations EXAMPLE 2, purely conceptual or syntheticallygenerated MVEs from statistics EXAMPLES 2-3 and 11, and with MVEs takendirectly from experimental images in a variety of ways, see, e.g.,EXAMPLES 5-6. An example of MVE construction and processing for metalsis given in FIG. 8, where a combination of statistical reconstructionand direct imaging is used to create an MVE.

For an example in composite materials, in the simplest form, the finalscale might be a component (scale 0, the largest) that is built withunidirectional carbon fiber reinforced composite (scale 1). At scale 1,a microstructure model to compute physical material responses isrequired. Once computed, this response is provided to scale 0 as theelement-level behavior. This is illustrated in FIG. 5. To capture itsresponse, the microstructure is modeled as a high-fidelity MVE. Each MVEcontains millions of voxel elements, as shown in FIG. 4. One mightenvision how this same scheme might be extended to scale 2, e.g., for awoven composite including unidirectional fiber in each yarn, as shown inFIG. 7. For arbitrary material systems, this scale decomposition mightcontinue to scale N.

Metals are another example of materials displaying functional andstructural hierarchy. In one simple case, metal might be represented bya homogenized behavior at scale 0, while scale 1 describes the behaviorat the granular level. The collective behavior of many grains works togenerate the homogenized response observed in scale 0, see, e.g.,EXAMPLE 2. Another might be that of defects (inclusions, voids) withineach grain, and yet another the scale of precipitates and dislocations.

According to the invention, the core of the method is composed of thefollow three steps, outlined as the top row in FIG. 1:

(1) Data collection using the high-fidelity MVE. This operation uses ahigh-fidelity approach with specifically crafted boundary conditions andproperties. This gives an indicator of the material response of the MVE,for each MVE. The data collected for each MVE are called “responsefields,” implying that there is not necessarily a unique choice ofmaterial response required. Often elastic response is desirable for itssimplicity, but the method can use any response field deemed appropriate(for example, plastic strain is another possibility). This data is savedin a database.

(2) Perform unsupervised learning on the response data obtained in step(1). This process marks all points within the MVE with a unique ID, suchas 1, 2, 3, etc. Thence the description of the MVE behavior can rely onthe groups of points with the same ID, rather than the pointsthemselves. We call each group a “cluster.” This is illustrated in FIG.2. The mapping between Cluster ID and spatial point is stored in adatabase called “clusters.” The clusters must be space-filling, thougheach cluster need not be contiguous.

(3) Compute the interaction tensor for all pairs of clusters (ratherthan points). This process computes pairwise interaction tensors for allclusters. This allows one to compute behavior of each cluster when someforcing term is applied on the MVE. At this step, the original MVE hasbeen completely replaced by the “compressed” MVE mathematicallydescribed with only clusters, not spatial points/elements.

Above three-step approach is depicted in further detail in FIG. 6. Afterthis has been done, the data is read in by the prediction operator inthe lower portion shown in FIG. 1. This approach leads to a drasticallyreduced computational cost for solving the MVE responses, as shown inTable 1, but the accuracy is preserved, as shown by an example in FIG.9. The computational time comparison for cases given in FIG. 9 isprovided in Table 1. With such a drastic reduction in computationaltime, practical modeling challenges requiring description of a materialat multiple scales are overcome using desktop-workstation-levelcomputational resources.

The reduction of complexity is illustrated in FIG. 73. The selection ofthe MVE size can be determined through a statistical analysis process.The interaction tensor computation can be accelerated using variousformulations. As shown in Table 1, one can see that the model withoutany data compression is big and the simulation is costly to perform. Theinvented method significantly reduces the overall system complexity(e.g., as measured by degrees of freedom in the solution) and thecomputational cost.

TABLE 1 Comparison of computational time and average percent differencebetween the DNS and reduced order MVE calculation. Difference MVE to DNSMVE Type Complexity Computational Time Solution High fidelity MVE36,000,000 200 hr. with 80 CPU cores 0% (DNS) voxels (57.6 MCPU-seconds) Reduced order 10 clusters Offline data collection, 8clusters: MVE 8,280 s 3.83% Online prediction, 2 s with 1 16 clusters:CPU 1.46%

The superior efficiency can also be used to build a complex materialmicrostructure response database using the reduced order model (ROM),e.g., EXAMPLE 1. One possible use for such a database is for training aphysics based neural network, which provides almost instantaneousmicrostructure responses under arbitrary external loading condition, asshown schematically in FIG. 10. The physics based neural network can beused for structure optimization and design, as suggested in EXAMPLE 1.Moreover, the methodology described is suitable for materials andsystems design for desired performance indices, as illustrated in FIG.11.

FIG. 3 shows example material systems with two or more relevant lengthscales that could be modeled by the method and system.

FIG. 4 shows one exemplary MVE model of a unidirectional carbon fiberreinforced composite containing 36 million voxels. On the left, the bluecylinders represent carbon fiber, and the red is a polymetric epoxyresin matrix materials. The overall (homogenized) response to an appliedload in the y-direction is plotted in the top right, and the localstress contours are plotted on the bottom right. The DNS process tocompute this single load history takes about 200 hours using 80 computerprocessors.

FIG. 5 is an illustration of the 2-scale multiscale problem and itscomplexity using the traditional finite element method. At themacroscale, a mesh of a generic part subject to some boundary conditionis shown. At each material point within these elements, a microscaleresponse corresponding to the behavior of a generic subscale, shown inthis example with spherical inclusions, is computed. Bottom:Illustration of two-scale multiscale clustering method and itscapability in order reduction. In this illustration, each colorrepresents order 1 degrees of freedom, rather than each blue-boundedregion within the top illustration.

FIG. 6 shows an exemplary three-step “training” process, as one possibleway to compute the clusters and interaction tensor for macroscalegeometry (top) and microscale geometry (bottom).

FIG. 7 shows an exemplary three-scale FRP modeling framework. Themacroscale model is a woven laminate composite model, built as a FiniteElement mesh. Each integration point in the macroscale model isrepresented by a woven MVE, which can have different tow size, towspacing, and tow angle (the one shown is 90° tow angle). Eachintegration point in the mesoscale model is represented by a UD MVE,which can have different fiber orientation, fiber volume fraction, andmatrix-fiber interfacial strength. If all three scales are discretizedwith FE meshes, the total DOF for this full-field system could be3.3×10¹⁵. Using the present method and replacing the mesoscale and themicroscale with ROMs, the total DOF is reduced down to 1.6×10⁶, areduction of nine order of magnitude.

FIG. 8 shows that in certain embodiment, MVEs might be generated from anumber of sources; this example for metallic materials shows grainsmeasured using x-ray diffraction, reconstructed from a statisticaldescription, and predicted from a processing model. Defects, voids inthis case, can be from measurements or predictions, here twoexperimental methods (x-ray tomography and FIB-SEM serial sectioning)are given as examples. These input data are paired (spatially), and usedfor the MVE, resulting in response predictions based directly onexperimental images.

In FIG. 9, left panel is the original DNS description of a UD MVE,center panel is cluster-based description (showing 2 clusters in thefiber phase, 8 in the matrix), visualized on the underlying voxel mesh,and right panel is numerical homogenization results for overallstress/strain. The DNS reference solution is shown in red with pointwise5% error bars. Cf. the reduced order solutions with 8 clusters (blue)and 16 clusters (green).

Referring to FIG. 10, a physics guided NN may be “layered” on top of theMVE ROM: the NN is trained on a large database of rapidly-computedbehavior. Once trained, this NN is thought to contain microstructuralinformation similar to the ROM, but is much faster to evaluate. Thisalternative makes it practical to conduct microstructure-basedstructural optimization and design.

FIG. 11 shows an application example: a composite design. The compositestructure can be designed with different microscale structure (e.g.,fiber shape) and mesoscale structure (e.g., fiber orientation in eachply, fiber orientation and fiber shape in each ply, as well as differentweave pattern for woven composite). Key performance indices (e.g.,strength and maximum deformation under external load) for the compositestructure can be predicted. If those indices do not meet the desiredcriterion, optimization routine will be called to update microscale andmesoscale structure in order to improve performance indices.

According to the invention, the process is fast, see, e.g., EXAMPLES5-8, among others, as shown in Table 1.

It is also efficient, based on the following justification (EXAMPLES5-8). The efficiency comes from the fact that the complexity of theproblem is reduced significantly after the data compression. The resultspresent in EXAMPLE 5, illustrate the saving of computational cost, asshown in Table 5-4. The present method provides considerable reductionin terms of computer memory needed and computational time needed,compared to the traditional approaches, such as FEM and FFT.

Further, the methos is also accurate, as demonstrated in Table 1 andshown in FIG. 9.

Moreover, the method can be applied to materials with one or more lengthscales, as provided in EXAMPLES 8 and 9.

In addition, the method can provide multiscale modeling capability forarbitrary number of scales. In EXAMPLE 9, the method has been applied towoven composite, which is made of yarn (with microstructure similar toUD composite illustrated in FIG. 7) and matrix materials. The wovenmicrostructure is given in FIG. 12, along with cluster distribution inthe matrix and yarn phase. The matrix is modeled with traditionalmaterial law, and the yarn is modeled by the Unidirectional (UD)composite microstructure to capture the realistic yarn materialmicrostructure. Since the woven composite is represented by clusters, itcan be applied to a higher scale woven laminated woven, depicted in FIG.7, and realized as a 3-scale composite model.

In one embodiment, the woven microstructure database is used to performa woven shear simulation, the loading direction is given in FIG. 167.The simulation is a three-scale model, where the macroscopic model is asingle element model in a finite element framework (FEF). A 2-scalewoven microstructure database, whose yarn phase is modeled as ahomogenized anisotropic elastic material, is also tested.

As shown in FIG. 167, the 3-scale woven provides highly nonlinear wovenresponses due to the yarn elasto-plastic behavior. The 3-scale model isable to capture such non-linearity due to yarn plasticity, which cannotbe captured using a simplified 2-scale model with elastic yarnproperties. FIG. 167 Woven RVE shear loading simulation by a singleelement Finite Element model. The results from 3-scale woven model and2-scale woven model are plotted.

According to the invention, the method can also utilize arbitrary shapeswithin fixed bounds shown in EXAMPLES 8 and 9. As shown in EXAMPLE 9,the method can be applied to both fiber reinforced composite and wovencomposite, where the fiber reinforce composite MVE has been illustratedin FIG. 9 and the woven composite MVE is illustrated in FIG. 12.

The method can be applied to arbitrary number of phases/constituents(EXAMPLES 4-5). In certain embodiments, the method has been applied to2-phase and 3-phase filled rubber composite, as illustrated in EXAMPLE 5and FIGS. 83 and 86. Referring to FIG. 83, domain decomposition of2-phase filled rubber into a reduced order model is represented by 64clusters. Left panel is an original voxel mesh for the 2-phase filledrubber; and right panel is a compressed 2-phase filled rubber model with32 clusters in the matrix phase and 32 clusters in the filler phase.

FIG. 13 shows a random grain structure with 35 grains (panel (a)), whereeach grain is considered a different material phase, and grain-by-grainstress predictions, with progressively more clusters per grain (panel(b)), as disclosed in EXAMPLE 4.

The method is predictive (solutions provided outside the bounds of knowndata), see EXAMPLES 3 and 9. In a certain embodiment, the method isimplemented for concurrent modeling for UD Carbon Fiber ReinforcedPolymer (CFRP) as described in EXAMPLES 4 and 11. The predictivecapability is illustrated in FIGS. 11, 12, 83, 86, and 167, see EXAMPLES3 and 9.

Additionally, the method is also descriptive (provides homogenized andfull field information), as shown in EXAMPLES 3 and 8.

Furthermore, the method could be implemented in hierarchical modelingschemes, see EXAMPLE 6. The method could also be implemented forconcurrent modeling schemes, as discussed in EXAMPLES 3 and 8, in whichseveral case studies for concurrent capture of macroscale physical fieldevolution and microscale physical field evolution are presented, and forcombined hierarchical and concurrent modeling schemes (EXAMPLE 7), asexplained in FIG. 14, which is an illustration of an implementation withhierarchical modeling between mesoscale and microscale, and concurrentmodeling between macroscale and mesoscale. The UD MVE elastic propertiesare pre-computed and passed to yarn, constituting to a hierarchicalmodeling process. When the woven MVE is under external loading, itsresponse is computed using both yarn (given by the UD MVE) and matrixproperties. When the FE model is under external loading, its localresponses is computed using the woven MVE. The FE model and woven MVEresponses are computed in a concurrent fashion, establishing aconcurrent modeling scheme. A combined hierarchical and concurrentmodeling is thus implemented.

Other applications of the method include materials design (materialphase selection), as illustrated in FIGS. 4, 9, and 14, where thematerial properties of each phase will change the MVE behaviors. Forexample, as shown in FIG. 4, different fiber and matrix properties willresult in different stress and strain curves, and the prediction can bemade within seconds. This provides the opportunity to generate a largematerial response database so the optimal material properties of eachphase can be selected.

EXAMPLES 3 and 8 disclose embodiments of the invention formicrostructure design. In addition, the method can be used with anyunderlying data representation: images, particles, meshfree, finiteelement meshes, etc., see EXAMPLES 2 and 5-6. The case shown in FIG. 83utilizes a voxel mesh directly generated from the 3D TEM process, seeEXAMPLE 5.

In one embodiment, the method can be used with machine learning forexpedited efficiency (GPU and different NNs), e.g., EXAMPLE 1. Themethod can also be used to create a microstructure response databasethat contains many pairs of stress and strain states. The database canthen be used as an input to the supervised learning algorithm, one typeof machine learning method, to training a Feed Forward Neural Network(FFNN) that can further improve the efficiency. The FFNN can predictstress state given arbitrary strain state that is within themicrostructure response database. As shown in Table 1-5 a speed-up of10000 can be achieved with FFNN.

In certain embodiments, data-driven materials and structure design areachieved with machine learning techniques. As presented in EXAMPLE 1, amicrostructure-based optimization is formulated. It allows to perform atopology optimization to design a structure with microstructure damageinformation, as shown in FIGS. 12-15, 58, 83, 86 and 167.

Process-structure-property-performance for different processes (AdditiveManufacturing (AM), composite, polymer matrix composite (PMC)), leadingto system design including experiments to predictions can be found inEXAMPLE 6.

In addition, multiscale structure-property materials and structuresdesign are illustrated by two examples (composites and alloys) in FIGS.4 and 7. As shown in FIG. 7, the method can be used for a 3-scalematerial system, and can be extended to N-scale.

For composite materials, such as woven composites, the method can beused for fast prediction of the overall material yield surface, asdescribed in EXAMPLE 7. The method can create a ROM for the woven MVE,which provides efficient prediction of woven composite elasto-plasticmaterial responses. As shown in panel (a) of FIG. 15, Table 4 andEXAMPLE 7, yield stresses are computed within one minute, providingconsiderable savings compared to experimental approaches. The 3D yieldsurface is shown in panel (b) of FIG. 15, and the yield surface can beused to determine if an applied stress state results in yielding of thematerial.

In addition, the method can be used to generate a vast woven compositeyield surfaces based on different microstructure and material propertiesof yarn and matrix, providing an enlarged material design space whileminimizing computational cost.

TABLE 4 Six yield points used to calibrate the yield surface as shown inEXAMPLE 7 Yield stress components (GPa) σ_(xx) σ_(yy) σ_(z) σ_(yz)σ_(xz) σ_(xy) Yield point 1 0.093 0 0 0 0 0 Yield point 2 0 0.093 0 0 00 Yield point 3 0 0 0.076 0 0 0 Yield point 4 0 0 0 0.031 0 0 Yieldpoint 5 0 0 0 0 0.031 0 Yield point 6 0 0 0 0 0 0.041

Without intent to limit the scope of the invention, examples accordingto the embodiments of the present invention are given below. Note thattitles or subtitles may be used in the examples for convenience of areader, which in no way should limit the scope of the invention.Moreover, certain theories are proposed and disclosed herein; however,in no way they, whether they are right or wrong, should limit the scopeof the invention so long as the invention is practiced according to theinvention without regard for any particular theory or scheme of action.

Example 1 Clustering Discretization Methods for Generation of MaterialPerformance Databases in Machine Learning and Design Optimization

Mechanical science and engineering can use machine learning. However,data sets have remained relatively scarce; fortunately, known governingequations can supplement these data. This exemplary study summarizes andgeneralizes three reduced order methods: self-consistent clusteringanalysis, virtual clustering analysis, and FEM-clustering analysis.These approaches have two-stage structures: unsupervised learningfacilitates model complexity reduction and mechanistic equations providepredictions. These predictions define databases appropriate for trainingneural networks. The feed forward neural network solves forwardproblems, e.g., replacing constitutive laws or homogenization routines.The convolutional neural network solves inverse problems or is aclassifier, e.g., extracting boundary conditions or determining ifdamage occurs. In this example, we explain how these networks areapplied, then provide a practical exercise: topology optimization of astructure (a) with non-linear elastic material behavior and (b) under amicrostructural damage constraint. This results inmicrostructure-sensitive designs with computational effort only slightlymore than for a conventional linear elastic analysis.

Computational methods in materials mechanics have evolved with thedevelopment of computation tools. A recent advance in computer sciencesis the development of the so-called “Big Data” era, where a combinedexplosion in the number of sensors and datapoints along sidecomputational resources and methods have enabled tracking and usinglarge databases to develop understanding of the world, often replacingsmaller more targeted studies that may produce less generalizableresults or lack key insights. Taken in the context of computationalmechanics, we can develop data-driven computational tools that rely onvast amounts of background data to facilitate, e.g., real-timemultiscale simulations for fast multistage material system design,in-the-loop mechanics for controls (e.g., in manufacturing).

In order to develop such data-driven computational tools, two primaryareas of study have emerged: (1) generation of materials systemdatabases for materials mechanics, typically using data compression, toreduce computational complexity; and (2) utilization of the database forreal-time response prediction and multi-stage design.

The first area arises because data science relies heavily on the sizeand reliability of the database available. Unlike traditionalapplications in data science, such as image detection/recognition orautomatic control, extremely large and well defined dataset aregenerally unavailable, or merely inaccessible, in computationalmechanics. Whether databases are constructed from experiments orcomputational models, the cost to generate data at a scale used in,e.g., image recognition has thus far been largely insurmountable. Oneapproach to this challenge has been to use multiscale simulations ofmaterial systems using fast calculations of the overall stress of arepresentative volume element (RVE) for arbitrary far-field deformationloading. Many methods have been developed with the goal of finding anappropriate balance between cost and accuracy for such a problem; theseare generally referred to as reduced order methods (ROMs), and many havebeen developed. In certain embodiments, RVE is a specific case of theMVE which will result in a converged solution of the microstructure itrepresented. The method can be applied to a RVE, but also to a MVE tocover a wide range of material systems. The second area has often beenaddressed with methods from machine learning and neural networks toprovide real-time prediction and multi-stage design. Data-driven methodshave also been used to enhance computational mechanics by, for example,optimizing numerical quadrature and replacing empirical constitutivelaws with experiment data. Recently, a deep material network method wasproposed which mimics neural networks topologically to link the micromaterial stiffness to the macro material stiffness. Once trained on apre-simulated micro-macro stiffness database it can be used to compute,with significant speed-up, the overall stress of an RVE under arbitraryfar-field deformation loading.

In this exemplary study, three techniques in modeling microstructurebased on data mining are explored and generalized first. These kinds offast methods address the first area: they can be used to generate thetype of very large databases required for the pure or mechanics-enhancedmachine learning. The workings of two different classes of neuralnetworks are then derived. Next, an engineering application for thesenetworks, topology optimization considering microstructure, is exploredwith detailed examples. Then some possible future directions fordata-driven computational approaches are outlined to inspire furtherresearch in this emerging field within computational mechanics.

Two-Stage Clustering Analysis Methods

The Self-consistent Clustering Analysis (SCA) and its close relativesVirtual Clustering Analysis (VCA) and FEM Clustering Analysis (FCA) aretwo-stage reduced order modeling approaches including an offline datacompression process and an online prediction process. This is conciselyillustrated in FIG. 16. In the offline stage, the original high-fidelityRVE represented by voxels or elements is compressed into clusters. Inthe online stage, macroscopic loading is applied to the reduced order(clustered) RVE. The system of equations describing mechanical responseis then solved on only the reduced representation as a boundary valueproblem. The notation used in this section is summarized in Table 1-1.

TABLE 1-1 Notation used for two-stage clustering analysis methods XMaterial point X′ Any other material point n Normal to boundary σ^(M)Macroscale stress tensor εM Macroscale strain tensor σ(X) Microscalestress tensor ε(X) Microscale strain tensor s Unit eigenstress e Uniteigenstrain {tilde over (Γ)}(X, X′) Green's operator {tilde over (Γ)}⁰Fourier transform of the periodic Green's operator I Counting index forclusters J Another counting index for clusters χ^(I)(X) Characteristicfunction for cluster I D^(IJ) Interaction tensor for discretizedLippmann- Schwinger equation B “Interaction tensor” for FCA Ω Materialdomain Ω^(I) Domain of the I^(th) cluster (subset of) c¹ Volume fractionof the I^(th) cluster {tilde over (ε)} Reference material strain {tildeover (C)} Reference material stiffness {tilde over (S)} Compliancematrix for reference material A Strain concentration tensor F(X)Deformation gradient at point X F⁰ Reference deformation gradient A′Deformation concentration tensor λ⁰, μ⁰ Lame's constants of thehomogeneous stiffness tensor ξ Fourier point

,

⁻¹ Forward and inverse fast Fourier transform techniques Δ▪ Incrementalform of an arbitrary variable ▪ r¹ Residual of the I^(th) cluster MJacobiam matrix of r with respect to Δε I₄ fourth-order identity tensorC_(alg) ^(J) Tangent stiffness of the material in the J^(th) clusterN_(C) Number of clusters N_(F) Number of Fourier points N_(I) Number ofintegration points N_(E) Number of finite elements E_(matrix) Young'sModulus of the matrix ν_(matrix) Poisson's Ratio of the matrixE_(inclusion) Young's Modulus of the inclusion ν_(inclusion) Poisson'sRatio of the inclusion σ_(Y,matrix) Yield strength of the matrix ε ^(p)Effective plastic strain (matrix)

To define the boundary value problem, consider a material occupying Ω⊂

^(d). The goal of homogenization is to find the macroscopic constitutiverelation between a macroscopic stress

$\begin{matrix}{{\sigma^{M} = {\frac{1}{\Omega }{\int{{\sigma(X)}{dX}}}}},{X \in \Omega}} & \left( {1\text{-}1} \right)\end{matrix}$

and a macroscopic strain

$\begin{matrix}{{ɛ^{M} = {\frac{1}{\Omega }{\int{{ɛ(X)}{dX}}}}},{X \in \Omega},} & \left( {1\text{-}2} \right)\end{matrix}$

where |Ω| is the total volume of the region. We define mathematicallythe RVE problem as

$\begin{matrix}\left( \begin{matrix}{{{\nabla{\cdot \sigma}} = 0},{\forall{X \in}}} \\{{ɛ = {\frac{1}{2}\left( {{\nabla u} + {u\nabla}} \right)}},{\forall{X \in \Omega}}} \\{{\sigma = {\sigma\left( {ɛ;X} \right)}},{\forall{X \in}}} \\{{{Boundary}\mspace{14mu}{conditions}},}\end{matrix} \right. & \left( {1\text{-}3} \right)\end{matrix}$

where σ=σ(ε; X) is a general microscale constitutive law. For thehomogenization problem, boundary conditions have to be chosen to satisfythe Hill-Mandel condition. In this exemplary study, the periodicboundary conditions are used: ε periodic and σ·n anti-periodic on ∂.

Continuous and Discretized Lippmann-Schwinger Equation

By introducing a reference material with distributed elastic stiffness{tilde over (C)}(X), it has been shown that the RVE problem withperiodic boundary conditions is equivalent to the integral equation

ε(X)={tilde over (ε)}(X)−∫{tilde over (Γ)}(X,X′):(σ(X′)−{tilde over(C)}(X′):ε(X′))dX′  (1-4)

where {tilde over (ε)}(X) is the strain in the reference material whenapplying the same loading and boundary conditions as the original RVEproblem; σ(X′)−{tilde over (ε)}(X):ε(X′) is the eigenstress applied tothe reference material; {tilde over (Γ)}(X,X′) is the Green's operatorassociated with the reference material. The physical meaning of −{tildeover (Γ)}_(ijkl)(X,X′) is the strain component ε_(ij) at material pointX if the unit eigenstress s^(kl) is applied at material point X′, withthe components of s^(kl) defined by s_(mn) ^(kl)=δ_(km)δ_(ln), whereδ_(km) and δ_(ln) are Kronecker delta functions.

The integral equation given in Eq. (1-4) is known as theLippmann-Schwinger equation, commonly seen to describe particlescattering in quantum mechanics. There is typically no explicit form of{tilde over (Γ)}(X,X′), unless the reference material is homogeneous.

The domain can be decomposed into several sub-regions, called clusters,distinguished mathematically by the characteristic function x as shownin Eq. (1-5).

$\begin{matrix}{{\chi^{I}(X)} = \left( \begin{matrix}{1,} & {X \in \Omega^{I}} \\{0,} & {otherwise}^{\prime}\end{matrix} \right.} & \left( {1\text{-}5} \right)\end{matrix}$

where I=1, 2, 3, N_(C) denotes each cluster, and Ω^(I) is the subset ofthe volume within cluster I. These clusters are defined during theoffline stage. This allows one to discretize the Lippmann-Schwingerequation, as:

ε^(I)={tilde over (ε)}^(I)−Σ_(J=1) ^(N) ^(C) D ^(IJ):(σ^(J) −{tilde over(C)} ^(J):ε^(J)),∀I∈{1, . . . ,N _(C)}  (1-6)

where

$\bullet^{I} = {\frac{1}{c^{I}{\Omega }}{\int{{\chi^{I}(X)}{\bullet(X)}{dX}}}}$

denotes the volume average of an arbitrary variable ▪ in the Ithcluster;

$c^{I} = \frac{^{I}}{\Omega }$

is the volume fraction of the Ith cluster where the volume of the Ithcluster is given by |Ω^(I)|. D^(IJ) is the interaction tensor given by

$\begin{matrix}{D^{IJ} = {\frac{1}{c^{I}{\Omega }}{\int{\int{{\chi^{I}(X)}{\chi^{J}\left( X^{\prime} \right)}{\overset{\sim}{\Gamma}\left( {X,X^{\prime}} \right)}{{dXdX}^{\prime}.}}}}}} & \left( {1\text{-}7} \right)\end{matrix}$

The physical meaning of −(D_(ijkl) ^(IJ)) is the average straincomponent ij in the Ith cluster if the uniform unit eigenstresscomponent kl is applied in the Jth cluster.

Remark 1: If homogeneous reference material is used, as in SCA and VCA,{tilde over (ε)}^(I)=ε^(M).

Remark 2: A counterpart of Eq. (1-6), similar to the idea of FCA, can beexpressed as

σ^(I)={tilde over (σ)}^(I)−Σ_(J=1) ^(N) ^(C) B ^(IJ):(ε^(J) −{tilde over(S)} ^(J):σ^(J)),  (1-8)

where {tilde over (S)} is the compliance matrix of the referencematerial; a is the stress in the reference material when applying thesame loading and boundary conditions as the original RVE problem.ε^(J)−{tilde over (S)}^(J):σ^(J) is the volume average eigenstrain inthe Jth cluster. The physical meaning of −B_(ijkl) ^(IJ) is the averagestress component ij in the Ith cluster if the uniform unit eigenstraincomponent kl is applied in the Jth cluster. Note that the referencematerial for FCA is the elastic state of the original RVE, instead of ahomogeneous reference material as used by the other two methods here.

Remark 3: The relationship between B^(IJ) and D^(IJ) is given by

{tilde over (S)} ^(I) :B ^(IJ) =−D ^(IJ) :{tilde over (C)} ^(J).  (1-9)

This can be proven by noting that the effect of applying any uniteigenstrain kl in some cluster J of the reference material is equivalentto that of applying eigenstress −{tilde over (C)}:e^(kl) in the samecluster.

In the online stage, SCA solves the incremental, discretizedLippmann-Schwinger equation, Eq. (1-6), with arbitrary external loadingconditions. These can either be of the fixed strain increment ε^(M) typeor be of the fixed stress increment σ^(M) type.

Offline: Clustering and Interaction Tensor

The offline stage includes three primary steps: (1) data collection, (2)unsupervised learning (e.g., clustering), and (3) pre-computation of theinteraction of clusters. Computation of the linear elastic response(data collection) and subsequent clustering based on that response areconducted identically for each of the three two-stage methods presentedbelow: the same voxel mesh and clusters are used in the examples for allthree methods. The difference between methods comes in the computationof the interaction tensor.

(1) Data Collection

Data collection provides information used to construct the reducedrepresentation of the system. It typically involves some computation ofthe response of a fully resolved system, perhaps with a simplifiedmaterial model, over a limited set of loading cases. One measure ofmechanical response that could be collected is the strain concentrationtensor used in micromechanics A, defined by ε(X)=A(X):ε^(M), which mapsbetween the far-field or applied strain, ε^(M) and the strain measuredat point X in the domain, ε(X).

The choice of the strain concentration tensor as given above is oftensuitable, but it depends on the relevant details of the problem. Forexample, at finite deformations one might consider using the deformationconcentration tensor:

${A^{\prime}(X)} = \frac{\partial{F(X)}}{\partial F^{0}}$

where F(X) is the deformation gradient at any given point X within thedomain and F⁰ is the macroscopic deformation corresponding to theboundary conditions. Alternatively, if the elastic and plastic materialresponses differ substantially, e.g., if one is isotropic and the otheranisotropic, including information about the plastic part of thedeformation might be desirable.

(2) Clustering

The goal of clustering is to reduce the number of degrees of freedomrequired to represent the system while minimizing the loss ofinformation about the mechanical response. One way of doing this is bygrouping material points within the domain of interest. If one canassume that the material response within each group is identical, theevolution of the domain can be determined by solving for the response ofeach group rather than each material point.

Using the data generated during the collection phase, one of the manyclustering (or unsupervised learning) techniques might be applied tooptimize the domain decomposition. In the following examples,Self-Organizing Maps (SOMs) are employed, as illustrated in FIG. 16,where eight clusters (four in each phase) are constructed. Theclustering process assigns each material point with a cluster ID, suchthat clusters are labelled 1, 2, N_(c).

The k-means clustering method has also been used. More elaborateclustering schemes might also be considered. Ongoing efforts include“adaptive clustering” schemes that mimic adaptive FE methods in theirability to evolve as deformation progresses, and “enriched” machinelearning, whereby a priori information from mechanics about thedeformation fields outside the bounds of the data collected in Step (1)are used to guide or bound the unsupervised learning. Another unexploredfuture direction might use a “feature-based” machine learning thatincludes microstructure information in addition to mechanicsinformation.

(3) Interaction Tensor

The interaction tensor describes the impact each cluster has on each ofthe other clusters. Once the clustering process is completed, theinteraction tensor can be explicitly computed. Importantly, the integralpart only has to be computed once during the offline stage. Only theresults of that calculation are then used for the online stage. Threeways to compute the interaction tensor, one used by each of the methodshighlighted here (though these are not exclusive to each), are:

A. SCA: Fourier Transform for D^(IJ)

With periodic boundary conditions and homogeneous reference material,the Green's operator has a simple expression in Fourier space, given by

$\begin{matrix}{{{\hat{\Gamma}}_{ijkl}^{0}(\xi)} = {\frac{\delta_{ik}\xi_{j}\xi_{l}}{2\mu^{0}{\xi }^{2}} - {\frac{\lambda^{0}}{2{\mu^{0}\left( {\lambda^{0} + {2\mu^{0}}} \right)}}\frac{\xi_{i}\xi_{j}\xi_{k}\xi_{l}}{{\xi }^{4}}}}} & \left( {1\text{-}10} \right)\end{matrix}$

where {circumflex over (Γ)}_(ijkl) ⁰=

(Γ⁰) is the Fourier transform of a periodic Green's operator Γ⁰; λ⁰ andμ⁰ are the Lamé's constants of the homogeneous stiffness tensor; is theFourier point. Then the interaction tensor can be calculated with

$\begin{matrix}{{D^{IJ} = {\frac{1}{c^{I}{\Omega }}{\int{{\chi^{I}(X)}{\mathcal{F}^{- 1}\left( {{\mathcal{F}\left( \chi^{J} \right)}{\mathcal{F}\left( \Gamma^{0} \right)}} \right)}dX}}}},{\forall I},{J \in \left\{ {1,\ldots\mspace{14mu},N_{C}} \right\}}} & \left( {1\text{-}11} \right)\end{matrix}$

using the fast Fourier transform (FFT) technique. The computationalcomplexity is O((N_(C))²(N_(F))log(N_(F))), where N_(F) is the number ofFourier points used in the FFT calculation.

B. VCA: Numerical Integration for D^(IJ)

With an infinite homogeneous reference material, the Green's operatorcan be expressed in real space. Numerical integration is the moststraightforward method to compute the integral equation given in Eq.(1-7). The computational complexity is O((N)²), where N_(I) is thenumber of integration points used.

C. FCA: Finite Element Method for B^(IJ)

Based on the physical interpretation of the interaction tensor, thefinite element method can also be used. By applying uniform uniteigenstrain component kl in the Jth cluster, the average stress can becomputed for all clusters, resulting in B_(ijkl) ^(IJ) for all I=1, . .. , N_(C). Thus, the computational complexity is O(6(N_(C))(N_(E))),where N_(E) is the number of finite elements used. The tensor B^(IJ) issimilar to the interaction tensor D^(IJ), although B^(IJ) is determinedby applying strains rather than stresses.

Online: Reduced Order Response Prediction

Once the interaction tensor database is prepared, the discretizedLippmann-Schwinger equation defined in Eq. (1-6) is solved in the onlinestage. An incremental form of Eq. (1-6) is given by

ε^(I)={tilde over (ε)}^(I)−Σ_(J=1) ^(N) ^(C) D ^(IJ):(σ^(J) :−{tildeover (C)} ^(J):ε^(J)),∀I∈{1, . . . ,N _(C)},  (1-12)

where {tilde over (ε)}^(I) is the applied incremental reference strain.The incremental stress σ^(I) is a function of the incremental strainε^(I) according to the local material constitutive laws. So the unknownsof Eq. (1-12) are the strains in each cluster {ε}={ε¹, . . . , ε^(N)^(C) }. For nonlinear microscale constitutive laws, Eq. (1-12) isnonlinear and has to be solved iteratively. Newton's iterative method isused by SCA and VCA, while a different iterative method is used by FCA.The residual form of Eq. (1-12) is given by

$\begin{matrix}{{r^{I} = {ɛ^{I} - {\overset{\sim}{ɛ}}^{I} + {\sum_{J = 1}^{N_{C}}{D^{IJ}:\left( {\sigma^{J} - {{\overset{\sim}{C}}^{J}:ɛ^{J}}} \right)}}}},{\forall{I \in \left\{ {1,\ldots\mspace{14mu},N_{C}} \right\}}}} & \left( {1\text{-}13} \right)\end{matrix}$

Then the Jacobian matrix

$\left\{ M \right\} = \frac{\partial\left\{ r \right\}}{\partial\left\{ ɛ \right\}}$

for the Newton's method is given by

$\begin{matrix}{{M^{IJ} = {\frac{\partial r^{I}}{\partial ɛ^{J}} = {{\delta_{IJ}I_{4}} + {D^{IJ}:\left( {C_{alg}^{J} - \overset{\sim}{C}} \right)}}}},{\forall I},{J \in \left\{ {1,\ldots\mspace{14mu},N_{C}} \right\}}} & \left( {1\text{-}14} \right)\end{matrix}$

where I₄ is the fourth-order identity tensor. The tangent stiffness ofthe material in the Jth cluster is C_(alg) ^(J). An alternative way tosolve for the local mechanical responses is to minimize thecomplementary energy of the clustering-based system.

Comparison of the Methods

In order to compare the accuracy and efficiency of each method, a 2Dplane strain model for a two-phase material is constructed and shown inFIG. 17. The 2D mesh contains 600×600 square pixels. The inclusion areafraction is 51%. Material constants of the matrix and the inclusion aregiven in Table 1-2. The yield surface is von Mises surface as shown inEq. (1-15). The hardening law of the matrix material is given in Eq.(1-16). Note that this will be approximated by a non-linear elasticbehavior for monotonic loading in future sections.

TABLE 1-2 Material constants for matrix and inclusion E_(matrix)E_(inclusion) Area fraction (MPa) ν_(matrix) (MPa) ν_(inclusion) ofinclusion 100.0 0.30 500.0 0.19 0.51

$\begin{matrix}{f = {{\overset{\_}{\sigma} - {\sigma_{Y,{matrix}}\left( {\overset{\_}{ɛ}}^{p} \right)}} \leq 0}} & \left( {1\text{-}15} \right) \\{\sigma_{Y,{matrix}} = \left( \begin{matrix}{0.50 + {5{\overset{\_}{ɛ}}^{p}}} & {0 < {\overset{\_}{ɛ}}^{p} \leq 0.04} \\{0.62 + {2{\overset{\_}{ɛ}}^{p}}} & {0.04 < {\overset{\_}{ɛ}}^{p}}\end{matrix} \right.} & \left( {1\text{-}16} \right)\end{matrix}$

The equivalent von Mises stress is σ. The yield stress σ_(Y,matrix) isgiven by the hardening law in Eq. (1-16) with equivalent plastic strainε ^(p). Strain from 0 to 0.05 is prescribed in the x-direction and zerostrain is enforced in the y- and xy-directions. A direct numericalsimulation (DNS) of the microstructure under these loading conditions isperformed using the FEM and the effective von Mises stress is recordedfor later comparison to the reduced order model results.

A one-time data-compression of the microstructure is performed using thestrain concentration tensor. The resulting clustering of themicrostructure is shown in FIG. 18. The interaction tensor for eachmethod is computed using the aforementioned algorithms.

The different methods, used to compute the interaction tensors, eachresults in a slightly different form of the tensor. There are strongsimilarities—after all, the same microstructure and clustering isused—though the details differ. FIG. 19 shows a magnitude plot of eachof the three methods, where magnitude represents the effect of eachstress component in the J^(th) cluster on the corresponding straincomponent in the I^(th) cluster. FIG. 19 shows component-wise magnitudeplots for D_(SCA) ^(IJ), D_(VCA) ^(IJ), B_(FCA) ^(IJ) in panels (a)-(c),respectively. Spikes along the diagonal direction for all threeinteraction tensor surface plots suggest self interaction has morecontribution than the rest of clusters in cluster-wise stress increment.D_(SCA) ^(IJ) and D_(VCA) ^(IJ) have the similar magnitude along theirdiagonal direction due to the homogeneous reference material assumption.B_(FCA) ^(IJ) has different magnitudes for matrix and inclusion phasealong the diagonal direction, implicitly representing a heterogeneousreference material. FIG. 20 is plots for D_(SCA) ^(IJ), D_(VCA) ^(IJ),B_(FCA) ^(IJ) in profile; note that for FCA the two regions correspondto different physical domains (matrix and inclusion). The trends of themagnitudes shown in FIG. 19 and FIG. 20 suggest that inter-clusterinteraction is not as strong for VCA as for SCA and FCA, although thisdifference is relatively minor. The strongest interactions areintra-cluster, as shown by the peaks along the diagonal. FCA has twodistinct regions of peaks, corresponding to the set of clusters in theinclusion and in the matrix. The tensor is constructed in an orderedway, which results in these two distinct sets of clusters. This isunlikely to change the overall solution accuracy.

Once the microstructural database is created and the interaction tensorhas been computed, the online prediction is performed. FIG. 21 showsthat the ROM results for stress in the x-direction in all three casesare within 5% of the DNS results, with small differences between thethree ROMs. Panel (b) of FIG. 21 highlights the slight differencesbetween the three methods: SCA follows the same trend as the DNS, but isslightly softer; the response predicted by VCA is softer still, as aresult of the constant reference material assumption; the trendexhibited by FCA is slightly different from the DNS overall, althoughstill quite close in value. FIG. 22, σ_(yy) ^(M) plotted against ε_(xx)^(M), shows that SCA has the best agreement with the DNS solution, andis the only prediction within 5% of the reference solution. Note thatVCA has different boundary conditions than the DNS solution, i.e., ituses a fictitious surrounding domain. Some deviation from a DNS resultwith periodic boundary conditions is therefore expected.

Machine Learning on Databases Generated with Predictive ROMs

Neural networks are a specific class of machine learning algorithms,which in the most basic form appear similar to regression analysis. Inpractice, these methods involve modifying input data through a series offunctions to obtain output data. The exact series of functions and theirweights and forms depend on the application. These methods can providean increase in speed over more conventional approaches, once thealgorithm has been appropriately trained. In order to have a highlyeffective modeling approach based on machine learning, rich databases ofmechanical response information are required to perform that training.Developing such a database with experiments is intractable, particularlyfor design of new materials or material systems where no materialperformance information exists. The fast, predictive models (SCA, VCA,FCA) outlined above are thus desirable for quickly populating relativelylarge materials databases. This enables the use of machine learningalgorithms in multiscale design, where simultaneously application andsatisfaction of criteria and constraints governing the materialmicrostructure and component-level macrostructure is required.

Feed forward neural networks (FFNNs) were the first neural networksdeveloped. The FFNN was designed to learn complex input-outputrelations. As such, FFNNs can be used to replace conventionalconstitutive laws; this is particularly appealing when descriptions ofthe homogenized behavior of a material is complex and/or difficult toobtain. The basic structure of an FFNN includes an input layer, hiddenlayers, and an output layer. Every pair of neurons in neighboring layershave a weighted connection. Each neuron in hidden layer and output layerhas a bias. In FFNNs, neurons in the same layer are not connected. Inthe learning procedure, the connection weights are changed following apredefined set of rules, such as with back propagation. Funahashi andHornik et al. proved that three hidden layers in an FFNN is sufficientto learn any non-linear continuous function.

Early efforts to apply machine learning to mechanics used a computationand knowledge representation paradigm, which is actually a type of feedforward neural network, to directly “learn” material behavior bytraining from analytic and experimental data. One early work applied aback-propagation neural network to model the behavior of concrete inplane stress under monotonic biaxial loading and compressive uniaxialcyclic loading. Once an RVE model is established, for example, an FFNNcan be trained on that data, and a concurrent multiscale scheme todirectly connect microstructure to the macro-scale material responsemight be achieved, with the FFNN replacing the RVE or constitutive lawto describe the response at each material point.

Neural networks have been studied with the goal of integration withmultiscale methods. In such cases, some parts of numerical simulationsare replaced with neural networks to better utilize their merits. Withthe development of numerical methods and the increasing interest inmultiscale simulation, integration of multiscale simulation and neuralnetworks is continuously developing.

In design optimization, one might expect there to be many calls to thematerial subroutine (e.g., one for each element for each load for eachdesign iteration). Thus running ROMs for RVEs might still be timeconsuming, compared to a model predicting RVE stress responses given astrain state within milliseconds, or a model providing strain stategiven microstructure stress contours within milliseconds. To tackle thisissue, we propose replacing the ROMs by neural networks trained on theRVE responses computed with SCA for micro-stress and macro-stress. Thesenetworks preserve microstructure information and are engineered to:

-   -   Predict macro (homogenized) stress or micro (local) stress given        a macro-strain using an FFNN for one RVE. In this case, the FFNN        plays the role of traditional material constitutive equations        and homogenization (to compute the macro-stress). We denote        these        _(FFFN) ^(micro) and        _(FFNN), respectively    -   Predict macro-strain given any micro-stress distributions using        a convolutional neural network (CNN). This is the inverse of the        FFNN. The input is a stress distribution within an RVE domain.        The output is the macro strain loading applied to this RVE (the        boundary conditions). We denote this        _(CNN).    -   Compute RVE damage by comparing local von Mises stress to a        stress threshold. A CNN is trained to identify the onset of        damage within an RVE. In this case, the CNN acts as a        classifier, which identifies whether or not the applied        macro-strain will cause microstructural damage. We denote this        _(CNN) ^(classify).

Database Generation for Machine Learning Using SCA

The ROMs presented in this example can be used to generate a databasefor machine learning. In this case, we are using SCA with 8 clusters. Ingeneral, such databases contain N_(T) training samples and N_(V)validation samples. For the RVE model, a database with N_(T)=1000strain-stress pairs is computed for a nonlinear elastic material byrandomly sampling 200 terminal states with four sub-loading steps each,as shown in FIG. 23. Similarly, N_(V)=150 strain-stress pairs (30 finalstates, plus four intermediate steps) are generated for validation. Amonotonically increasing load is assumed and all the final points areconfined to a spherical space with a radius of 0.05. Table 1-3 shows thereduction in time required to build a database achieved by using SCA,versus FEM or FFT for this example case. Without a reduced order model,the generation of microstructure database takes days using FEM or FFT.The database contains N_(T)=1000 pairs of macroscopic strains andstresses ε^(M,s), σ^(M,s), and local (micro)stress σ^(s)(X), where s=1,2, . . . , N_(T). In this case, we only consider a plane strain problemand do not consider stress in the z direction. SCA reduces the totalcomputational time by two orders of magnitude compared to FFT and byfour orders of magnitude compared to FEM.

FIG. 23 shows random samples of strain state; two hundred final stateswere selected, and four evenly spaced intermediate steps to reach thefinal states were recorded for a total of N_(T)=1,000 samples. Allstrain states will be applied to the RVE to generate correspondingstress states.

TABLE 1-3 Comparison of time required to generate a microstructuredatabase for nonlinear elastic RVE responses with N_(T) = 1,000 samplesof strain states. SCA requires only one single workstation to generatethe database. Speedup over Method Total time (s) FEM FEM 2.04 × 10⁷ —FFT 3.01 × 10⁵ 68 SCA offline: 13.0 + online: 9 400 (4 + 4 clusters)2.16 × 10³

Feed Forward Neural Networks

In order to illustrate the structure of feed forward neural networks(FFNNs), a simplified one dimensional example is presented. In linearelasticity, stress is related to strain by the material stiffness; thiscan be generically defined as a mapping. The overall structure of aneural network can also be described as a mapping, i.e.:

$\begin{matrix}\left( \begin{matrix}{{{Constitutive}\mspace{14mu}{{equation}:{Stress}}} = {\mathcal{F}_{Constit{utive}}({strain})}} \\{{{Neural}\mspace{14mu}{network}\mspace{14mu}{{mapping}:{Stress}}} = {\mathcal{F}_{F{FNN}}({strain})}}\end{matrix} \right. & \left( {1\text{-}17} \right)\end{matrix}$

where

_(FFNN) is the FFNN that uses strain state ε as input, and generatesstress state σ as the output. The structure of a simple FFNN is shown inpanel (a) of FIG. 24. As shown in FIG. 24, panel (a) is illustration ofan FFNN network with one hidden layer for a linear elastic example; thecollective function of the weights and biases connecting the input layer(green), the hidden layer (blue), and the output layer (red) is that ofYoung's Modulus E. panel (b) is s stress-strain diagram, showing how theinput strain is interpreted by the FFNN for a linear elastic case usinglinear activation functions and zero biases. The notation used in thisfigure and throughout this section is defined in Table 1-4. For thisillustration case, only one sample is considered, hence s=1 and allvariables are written without the superscript s. A general FFNN containsneurons (the circles) and weights (black lines). In general, an FFNN hasone input layer, one output layer, and multiple hidden layers. Eachlayer may have multiple neurons; for the input and output layers, theseare simple the input and output values. In the simplest case, an FFNNwould have one input neuron, one hidden neuron, and one output neuron.For 1D linear elastic stress analysis in such a case, the input neuronwould be strain, the hidden neuron would act as a multiplicative,functional decomposition of the stiffness that recovers the totalstiffness required to map strains to stress in the output neuron.Generalizing this slightly, we might consider an FFNN with three hiddenneurons, as shown in FIG. 24. Each neuron has only one value. The firstneuron is simply the strain:

a _(i=1) ^(l=1)=ε(input layer)  (1-18)

The three neurons in the hidden layer take in this value, and each takeon the value given by:

a _(j=1,2,3) ^(l=2)=

(Σ_(i=1) ¹ W _(ij) ^(l=2) a _(i) ^(l=1) +b _(j) ^(l=2))(hiddenlayer)  (1-19)

where

is an activation function. In the training part, this example usesSigmoid function:

${{f(x)} = \frac{1}{1 + e^{- x}}},$

and each neuron is computed using a different weight W_(ij) ^(l=2) andbias b_(j) ^(l=2), where i is the neuron in the previous layer (in thiscase, the input later) and j is the neuron in the current layer (in thiscase, the hidden layer). Finally, the overall response—the stress—isgiven by:

σ_(predicted) =a _(k=1) ^(l=3)=Σ_(j=1) ³ W _(jk) ^(l=3) a _(j) ^(l=2) +b_(k) ^(l=3) (output layer)  (1-20)

The combination of all the W and b terms, as well as the activationfunction, work as the fitting factors in a regression analysis. Theactivation function

is fixed for all neurons, and is used to condition the weightingfactors. For the constitutive model outlined above, the overall resultsof the weights, bias and activation function would perfectly match theelastic modulus. If we only consider weights and the activation functionis just a linear mapping, the function of each neuron in hidden layer isgiven explicitly in panel (b) of FIG. 24, where the bias is taken aszero for all neurons.

The physical interpretation of individual neurons is more complicatedfor non-linear responses, although the overall idea is the same. In thisformulation, strain path dependence (i.e., plasticity) is impossible tocapture, and the net result is a response map where one might think ofthe collection of neurons (weights, biases and activation functions) asan “instantaneous elastic modulus,” or the slope of a line that relatesstrain to stress at the current point in strain.

FIG. 25 is an illustration of an FFNN with multiple hidden layers;N_(L): index of layers, N_(N)(l): number of neurons in layer l. Theformulation of the FFNN is given in Eq. (9-21) with associatedinterpretation of the FFNN structure. The indices i and j represent theneuron ID in the previous layer and current layer, e.g., W₁₂ ^(l=2) isthe weight between neuron 1 in layer l=1 and neuron 2 in layer l=2. Theexample shown in FIG. 25 extends the previous example to consider atwo-dimensional stress analysis, with many neurons per layer and severallayers. The inputs in are three macroscale strain components ε_(xx)^(M), ε_(yy) ^(M), γ_(xy) ^(M), in a plane strain problem. The outputsare the three macroscale stress components ε_(xx) ^(M), σ_(yy) ^(M),τ_(xy) ^(M). The stress component σ_(zz) ^(M) is not considered. Thesamples (stress-strain pairs) from the database outlined above are usedto train the neural network. After training, the FFNN can predictstresses when given strain inputs.

In each layer of a general FFNN, each neuron takes the output value fromeach neuron in the preceding layer as inputs and gives a single output.This is repeated for each layer. Generalizing Eqs. (1-18), (1-19), and(1-20) to an arbitrary number of layers and neurons per layer results inEq. (1-21), where the value of the j^(th) neuron in layer l for thes^(th) sample (either a training sample or prediction) can be expressedas:

                                     (1-21)$a_{j}^{l,s} = \left( \begin{matrix}{ɛ_{j}^{M,s},{{ifl} = {1\left( {{input}\mspace{14mu}{layer}} \right)}}} \\{{\mathcal{A}\left( {{\sum_{i = 1}^{N_{N}{({l - 1})}}{W_{ij}^{l}a_{i}^{{l - 1},s}}} + b_{j}^{l}} \right)},{{ifl} \in {\left\{ {2,\ldots\mspace{14mu},{N_{L} - 1}} \right\}\left( {{hidden}\mspace{14mu}{layers}} \right)}}} \\{{{\sum_{i = 1}^{N_{N}{({l - 1})}}{W_{ij}^{l}a_{i}^{{l - 1},s}}} + b_{j}^{l}},{{ifl} = {N_{L}\left( {{outout}\mspace{14mu}{layer}} \right)}}}\end{matrix} \right.$

where the final layer gives the estimated stress:

σ_(prediceted,j) ^(M,s) =a _(j) ^(N) ^(L) ^(,s).  (1-22)

TABLE 1-4 Notation table of variables used in the feed forward neuralnetwork ε_(j) ^(M,s) Macroscale strain tensor, s = 1, . . . , N_(T) sCounting index for number of samples (training or validation, dependingon context) l Counting index for number of layers i Counting index forneurons in a given layer j Counting index for neurons in another layerN_(T) Number of training samples N_(L) Number of layers in the neuralnetwork N_(N)(l) Number of neurons in layer l W_(ij) ^(l) Weightconnecting the i^(th) neuron in layer l − 1 to the j^(th) in layer lb_(j) ^(l) Bias of the j^(th) neuron in layer l a_(j) ^(l,s) Neuronvalue for j^(th) neuron in l^(th) layer and for s^(th) sample

Activation function

_(FFNN) Feedforward neural network function

The outputs a_(j) ^(l,s) of each layer possesses similar physicalmeaning as explained for the 1D case. The input strain components arerepresented by l=1, a_(j) ^(l,s) for the s^(th) sample, and l=2, . . . ,N_(L)−1, a_(j) ^(l,s) represents an estimate of the the nonlinear stressresponses of the microstructure. The activation functions and weights oflayer l=2, . . . , N_(L)−1 play a roughly similar role to the classicdefinition of the tangent modulus in solid mechanics. The hidden layerstake in strain components and produce an estimate of the non-linearstress responses. During the training process, the non-linearrelationship between stress and strain is gradually “learned” by thosehidden layers. In the last layer, l=N_(L), a_(j) ^(l,s) represents thepredicted stress components. The predicted stress components areproduced through the regression operation in the output layer, as shownin Eq. (1-21). The weights and bias of the output layer correct theprediction generated from hidden layers, and produce accurate nonlinearstress responses. In order to make the concept clear, one might considerthe hidden layer as unitless values operating on intermediate strainvalues, while the units of W and b in output layer are those of stress(e.g., MPa in the example problem). The FFNN can learn nonlinear elasticmaterial behaviors due to following two key factors: 1) hidden layersapproximate the material nonlinear elastic responses as a traditionalconstitutive model would do, as described in Eq. (1-17); 2) the outputlayer corrects the predicted nonlinear responses for improved predictionaccuracy.

Feed Forward Neural Network with Database Generated by SCA

In this case study, an FFNN is trained with data generated using SCA.Using the same microstructure as given in FIG. 18, SCA computes thestress responses of the RVE (or any arbitrary RVEs) when a monotonicallyincreasing strain is applied. Since SCA provides efficient evaluation ofthe stress state, it is convenient to train the FFNN on data made withSCA. The FFNN can then replace SCA by “learning” the stress state as afunction of the strain state. This would establish a straight-forwardrelationship between strain and stress for near-instantaneous evaluationof RVE stress responses. Note that although a plastic material isdescribed for the matrix material in the RVE analyzed by SCA, the FFNNdescribed here is limited to non-linear elastic (i.e., path independent)material behavior: we approximate the plastic response with a non-linearelastic one, and focus on monotonic loading. Moreover, in the followingdesign case, the FFNN is trained to predict not only the overall RVEstress responses (as was shown in FIG. 25), but also cluster-wise, localstress responses. This is essentially the same process, but the outputlayer is size N_(C)×N_(N) (l=1), with one point for each stresscomponent for each cluster. This second form replicates thenon-homogenized results of SCA.

Training

The training procedure for an FFNN can be reformulated as anoptimization problem. We define the loss function (or cost function) asMean Square Error (MSE) for the estimated stress and stress computed byRVE using SCA. Assuming one hidden layer, the optimization formulationis given by:

$\begin{matrix}{\mspace{79mu}{{{{find}:W_{ij}^{l = 2}},b_{j}^{l = 2},W_{jk}^{l = 3},b_{k}^{l = 3}}{{{minimize}\mspace{14mu}{loss}\mspace{14mu}{{function}:{MSE}}} = {\frac{1}{N_{T} \times {N_{N}\left( {l = 3} \right)}}{\sum_{s = 1}^{N_{T}}{\sum_{k = 1}^{N_{N}{({l = 3})}}\left( {\sigma_{k}^{{l = 3},s} - \sigma_{k}^{*{,{l = 3},s}}} \right)^{2}}}}}{{{where}:\sigma_{k}^{{l = 3},s}} = {\sum_{j = 1}^{N_{N}{({l = 2})}}{W_{jk}^{l = 3}\left( {{\mathcal{A}\left( {{\sum_{i = 1}^{N_{N}{({l = 1})}}{W_{jk}^{l = 3}ɛ_{i}^{M,s}}} + b_{j}^{l = 2}} \right)} + b_{k}^{l = 3}} \right)}}}}} & \left( {1\text{-}23} \right)\end{matrix}$

By finding the optimal values for W_(ij) ^(l=2), b_(j) ^(l=2), W_(jk)^(l=3), and b_(k) ^(l=3), MSE is reduced. Note that only training datais used in this process, hence s=1, 2, . . . , N_(T).

Usually, the MSE gradually decreases with each training step. To ensurethe trained neural network is general enough for all possible inputstates, some data points called verification data are used to monitortrends in the error. The minimization iterations terminates before theerror of the verification data starts to increase. This ensures theneural network is able to provide certain extrapolating capability fordata points that are not within the training set.

The FFNN described above was trained on the database. In this case, anFFNN with one hidden layer and 50 neurons was chosen. In the trainingprocedure, 1,000 samples are used to train the neural network. TheLevenberg-Marquardt optimization algorithm is used to reduce the MSE.

Prediction

After the training process, a fast evaluation of the stress state duringa monotonic loading process was performed. The FFNN used for thispredicts the macroscale stress tensor for a given macroscale straintensor following Eq. (1-24); similarly, Eq. (1-25) is used to predictthe local (cluster-wise) stress tensors given a macroscale straintensor.

σ^(M,s(FFNN))=σ_(k) ^(l=N) ^(L) ^(,s)=

_(FNN)(ε_(i) ^(M,s))  (1-24)

σ^(s(FFNN))(X)=σ_(k) ^(l=N) ^(L) ^(,s)(X)=

_(FFNN) ^(micro)(ε_(i) ^(M,s))  (1-25)

To demonstrate validation of the macroscale FFNN, the l² norm is used tomeasure the difference between the overall stress predicted by Eq.(1-24) and the homogenized SCA results for each sample in the validationdata set, as computed by:

Difference_(FFNN) ^(s)=∥

_(FFNN)(ε_(i) ^(M,s))−σ^(M,s(SCA))∥₂ ·,s=1,2, . . . ,N _(V),  (1-26)

To validate the trained FFNN, another 30 final strain states (unknownduring the training) with five load steps each (including the finalstate) were selected. FIG. 26 shows a histogram of difference measuredwith the l²-norm, Eq. (1-26), for these new N_(V)=150 strain-stresspairs of in the validation data set. Most of the test samples have avery low l²-norm, which shows that the FFNN is well trained. FIG. 27shows the SCA stress data for each stress component plotted against theFFNN predictions; a perfect match has a slope of one. The associatedcross-correlation statistic is one: the FFNN solutions match the SCAsolution perfectly. At each load step the stress predictions of the FFNNand SCA match, as shown in FIG. 28.

This case study illustrates a convenient workflow that used the reducedorder modeling approach to generate a rich microstructure responsedatabase for training an FFNN, which is then used for generating fastpredictions of the RVE responses. Note that although the validation forthe FFNN for the homogenized stress-strain relationship is given indetail here, a similar process has been used for the relationshipbetween macroscale loading and microscale (cluster-wise, or local)stresses, as given in Eq. (1-25).

We propose that the FFNNs shown here can be used in a designoptimization process, such as topology optimization or microstructuraldesign, where a fast and accurate material responses prediction isdesired. However, note that the material is non-linear elastic and/orunder monotonic loading. If plasticity and loading/unloading areconsidered, a different FFNN setup or a different neural network may berequired. A speed comparison of running 150 samples with SCA and FFNN isgiven in Table 1-5, where the speedup for online prediction of σ^(M) is10000 for the FFNN over SCA. This idea will be explored further usingboth the FFNN predictions (Eq. (1-24) and Eq. (1-25)) and convolutionalneural networks.

TABLE 1-5 Comparison of time required to run 150 samples using SCA andFFNN. Online speedup Method Total time (s) over SCA SCA (4 + 4 offline:13.0 + — clusters) online: 3 × 10² FFNN training: 60 + 10 000prediction: 3 × 10⁻²

Convolutional Neural Network

Convolutional networks or convolutional neural networks (CNNs) arewidely used in fields such as image recognition and featureidentification. The term “convolutional” refers to the linearmathematical operation and indicates that the convolution operation isimplemented in at least one layer of the network rather thanconventional matrix multiplication. This is a biologically inspiredmodel used to handle known grid-like topology data such as time series(1D grid of samples at successive time intervals) or image data (2D gridof pixels). Convolution neural networks have been implemented inmaterial science and multiscale modeling to analyze the microstructureproperties where the input data are microstructure images. Extractingmaterial information through microstructure images, a ubiquitous datatype in materials science, has proven to be a promising application ofCNNs. For example, Lubbers et al. implemented a CNN based on thedistribution of texture images for unsupervised detection oflow-dimensional structures. Scanning electron microscope (SEM) imagesare frequently used in materials science to distinguish betweencategories of materials. Such image datasets can be classified with asingle feature or with multiply features using CNNs. Some studies haveimplemented CNN to featurize SEM images over a single set of data.However, a scalable and a generalizable feature should be used tofacilitate widespread applicability of the CNN. Ling et al. analyzed thegeneralizability and interpretability of CNN-based featurization methodsfor SEM images, and found that mean texture featurization is generallyuseful in such cases, although sometimes feature-free CNN procedures areappealing as well.

The application of CNNs is not limited to images. Cang et al.established a CNN approach to predict the physical properties of aheterogeneous material, replacing standard statistical ormicromechanical modeling techniques. The generated scheme is applicableto systems with a highly non-linear mapping based on high dimensionalmicrostructure. They have implemented a convolutional network toquantify material morphology followed by another convolution network topredict the material properties given the microstructure. This can alsobe done in 3D, for complex materials and responses.

A CNN model includes several basic unit operations: padding,convolution, pooling, and a feed forward neural network (FFNN). Thestructure of an example 1D CNN is shown in FIG. 29.

The input is a series of stress values, i.e., a 1D problem. The 1D CNNincludes several loops of padding, convolution, and pooling. For aspecific loop iteration η, a padding procedure adds zeros aroundboundaries, to ensure that the post-convolution dimension is the same asthe input dimension. After padding, several kernel functions will beused to approximate the discrete convolution operator given by:

{tilde over (σ)}_(x) ^(κ,η)=Σ_(ξ=−(L) _(conv) _(−1)/2) ^((L) ^(conv)^(−1)/2) w _(ξ) ^(κ,η)σ_(x+ξ) ^(padded,η) +b ^(κ,η,)  (1-27)

where σ_(x+ξ) ^(padded,η) is the input, w_(ξ) ^(κ,η) is the κ^(th)kernel function, and b^(κ,η) is the bias, for η^(th) convolution processand η=1, 2, . . . , N_(conv). The size of the kernel function isL_(conv). A summary of all of the notation used in this section is givenin Table 1-6. The convolution operation can be regarded as a featureidentification operation. After padding, a pooling layer will decreasethe dimension of inputs, and extract the most important features fromthe post-convolution data. A one dimensional max pooling equation isgiven by

{circumflex over (σ)}_(α) ^(max,κ,η)=MAX({tilde over (σ)}_(ξ)^(κ,η),ξ∈[(α−1)L _(pooling)+1,αL _(pooling)])

α=1,2, . . . ,N _(pooling) ^(η)  (1-28)

where {circumflex over (σ)}^(max,κ,η) is the output value, {tilde over(σ)}_(ξ) ^(κ,η) is the input value, and L_(pooling) is the length of thepooling window. Max pooling extracts the maximal value from the window,but other pooling operations might also be used. In this case we surmisethat, to predict remote strains, the maximum stress values might betelling. Padding, convolution, and pooling may be repeated for N_(conv)times. The value will be transferred to a fully connected FFNN, such asthat illustrated in the previous section.

FIG. 30 represents a generalized overview of the structure of a typicaltwo dimensional CNN. Continuing with the two-dimensional example problemgiven above, for a CNN the input is the stress within the 600×600 grid,given by σ(α,β) were α and β correspond to the x- and y-components ofthe stress map. At each grid point (α,β), the three stress componentsxx, yy and xy are stored. Padding

_(padding) will add zero boundaries around the input data to make surethat after convolution, the size of the maps will remain same.

Similar to 1D convolution, the convolution operation applies a kernelfunction over the stress contours and generates a new feature map withthe same resolution as the initial stress contours. The extension of theconvolution operation to two dimensions for η^(th) convolution processis shown in Eq. (1-29).

{tilde over (σ)}_(α,β) ^(κ,η)=Σ_(ζ=1) ^(N) ^(feature) (Σ_(ξ=−(X) _(conv)_(−1)/2) ^((X) ^(conv) ^(−1)/2)Σ_(ψ=−(Y) _(conv) _(−1)/2) ^((Y) ^(conv)^(−1)/2) W _(ξ,ψ) ^(ζ,κ,η)σ_(α+ξ,β+ψ) ^(padded,ζ,η))+b ^(κ,η)  (1-29)

where κ is the kernel ID of the convolution layer and goes from one toN_(kernel). The size of the convolution kernel in dimension 1 and 2 aregiven by X_(conv) and Y_(conv), and are both odd numbers. The countingindices in the kernel in dimension 1 and 2 are defined ξ as and ψ,respectively. The number of stress components is N_(feature); for this2D example, N_(feature)=3. By applying the kernel to each element ineach the input stress array, a complete feature map will be generated.

To define a nonlinear relationship between the input and output usingthe CNN, a nonlinear activation function is often used. In some cases,this is a Rectified Linear Unit (ReLU) layer, which is applied to allfeature maps generated from the convolution operation. For simplicity ofillustration, this step is not shown in the equations and figures.

A pooling layer is applied to all feature maps after the ReLU layer tocompress the resolution of the data in the X and Y directions. Differentpooling operations might be used; in this example, we selected maxpooling. The max pooling operator divides the feature map into manysubset regions, and selects the maximum value from each region to use asthe value in the compressed feature map; generalizing from Eq. (1-28),for the n^(th) convolution process, this can be written as:

{circumflex over (σ)}_(α,β) ^(max,κ,η)=MAX({tilde over (σ)}_(ξ,ψ)^(κ,η)ξ∈[(α−1)X _(pooling)+1,αX _(pooling)],ψ∈[(β−1)Y _(pooling)+1,βY_(pooling)])

α=1,2, . . . ,β=1,2, . . . ,N _(Y) _(pooling) ^(η)  (1-30)

After the final pooling operation, all compressed feature maps areconverted into a single vector through a flattening operation. Theflattened array is then used as the input of the FFNN for regression tocompute the corresponding strain. Further details of the CNN method andimplementation can be found in literature cited in the beginning of thissection.

TABLE 1-6 Notation used to describe the CNNs shown above. X A point inthe microscale (inside the RVE) region σ^(M) Macroscale stress tensorε^(M) Macroscale strain tensor σ(X) Microscale stress tensor ε(X)Microscale strain tensor α First direction in X β Second direction in X{tilde over (σ)} Microscale stress tensor to which a kernel has beenapplied {circumflex over (σ)} Microscale stress tensor to which a kerneland pooling has been applied N_(T) Number of training samples in thedatabase N_(V) Number of verification samples in the database N_(kernel)Number of kernels in convolution N_(pooling), N_(Xpoo) Size of outputafter pooling N_(fl) Number of entries in flattened vector N_(conv)Number of repeats of padding, convolution, and pooling in CNN W_(ij)^(l) Weight in FFNN connecting the i^(th) neuron in layer l − 1 to thej^(th) in layer l b_(j) ^(i) Bias in FFNN of the J^(th) neuron in layer1 ξ Counting index for location within kernel in dimension 1 ψ Countingindex for location within kernel in dimension 2 ζ Counting index forfeatures κ Counting index for kernels η Counting index for convolutionsW_(ξ,ψ) ^(ζ,κ,η) Weight connecting the input for ζ feature and κ kernelon convolution layer b^(κ,η) Bias of the kernel κ on the convolutionlayer L_(conv), X_(conv), Size of the kernel Y_(conv) L_(pooling), Sizeof the pooling window X_(pooling), Y_(pooling)

_(padding) Padding function

_(conv) Convolution function

_(pooling) Pooling function

_(pcp) Combined padding, convolution, and pooling operation

_(flatten) Flattening function

_(CNN) Convolutional neural network function

_(CNN) ^(classify) Classification convolutional neural network functiond Binary indicator given by classification CNNConvolutional Neural Network for Boundary Condition Identification withDatabase Generated by SCA

Using the CNN illustrated in FIG. 30, a mapping has been establishedbetween local stress distribution and applied external strain on themicrostructure.

Training

This CNN was trained on the same database of 1000 SCA results as was theFFNN. For the CNN, the input is the micro-scale stress at each point(voxel) of the RVE, given by the cluster-wise results of SCA, σ(α,β)Each point in the RVE contains the three 2D stress components, like theRGB channels used for images. The output of the CNN is the macroscalestrain ε^(M) that was applied as the loading conditions and caused theobserved stresses. The training can be written as an optimizationproblem, as given in Eq. (1-31). The equations for training a CNN withpadding, convolution, and pooling layers repeated N_(conv) times isgiven by:

$\begin{matrix}{\mspace{79mu}{{{{find}:W_{mn}^{l}},{b_{n}^{l}\left( {{l = 2},{3\mspace{14mu}\ldots\mspace{14mu} N_{L}}} \right)},{{in}\mspace{14mu}{FFNN}}}{W_{\zeta,\psi}^{\zeta,\kappa,\eta},{b^{\kappa,n}\left( {{\kappa = 1},{2\mspace{14mu}\ldots\mspace{14mu} N_{kernel}}} \right)},\left( {{\eta = 1},{2\mspace{14mu}\ldots\mspace{14mu} N_{conv}}} \right),{{in}\mspace{14mu}{CNN}}}\mspace{79mu}{{\min\mspace{14mu}{loss}\mspace{14mu}{{function}:{MSE}}} = {\frac{1}{N_{T}}{\sum_{s = 1}^{N_{T}}\left( {ɛ^{M,s} - ɛ^{{*M},s}} \right)^{2}}}}{{{where}:ɛ^{M,s}} = {\mathcal{F}_{FFNN}\left( {\mathcal{F}_{flatten}\left( {\mathcal{F}_{pcp}^{N_{conv}}\left( {\ldots\mspace{14mu}{\mathcal{F}_{pcp}^{2}\left( {\mathcal{F}_{pcp}^{1}\left( {\sigma^{s}\left( {a,\beta} \right)} \right)} \right)}\mspace{14mu}\ldots}\mspace{14mu} \right)} \right)} \right)}}}} & \left( {1\text{-}31} \right)\end{matrix}$

The term

_(pcp) ^(η)(σ^(s)(α,β)) is used to simplify the notation by combiningthe nested operations shown in FIG. 30. It is defined as

_(pcp) ^(η)(σ^(s)(α,β))=

_(pooling) ^(η)(

_(conv) ^(η)(

_(padding) ^(η)(σ^(s)(α,β)))),

which includes terms for padding, convolution, and pooling. Theremaining terms in the training problem are define as follows. Theweight and bias in the FFNN are W_(mn) ^(l) and b_(m) ^(l), and W_(ξ,ψ)^(ζ,κ,η), b^(κ,η) are the weights and biases in the convolutionoperations. The ground truth is ε^(+M,s) and the estimate is ε^(M,s).The number of training samples is defined as N_(T), and N_(N)(l=N_(L))is the number of neurons in output layer. In this case, N_(N)(l=N_(L))is three. The sampling is indexed by s.

The inputs to the CNN are the three stress arrays corresponding to thecomponents of stress given by σ^(s)(α,β). The outputs are three strainvalues ε_(j) ^(M,s)(j=1,2,3). Just as for the FFNN, the mean squarederror (MSE) is reduced gradually step by step using one of severaloptimization algorithms.

Prediction

Just as with the FFNN, we can define the function

_(CNN)(σ^(s)(α,β)) that describes the operation performed by the trainedCNN, in this case

_(CNN)(σ^(s)(α,β))=ε^(M,s)  (1-32)

In FIG. 31, a histogram for the error between predicted ε^(M) and thevalidation data set (again, the N_(T)=150 samples generated above) iscomputed using Eq. (1-33). Most of the predictions made by CNN have anl² norm less than 1×10⁻⁵, showing the CNN produce an accurate predictionof the strain state. The l² norm illustrate the CNN network is able tomake a proper prediction of the validation data. In FIG. 32, thecorrelations between the CNN prediction of applied strain and thereference solution of the three strain components is provided, using thesame validation data sets as mentioned before. The solid black lines arethe ground truth: all perfect predictions should lay on those lines. Allthree cases have correlation coefficients higher than 0.99, suggestingthe trained CNN can provide a good accuracy in predicting appliedstrain. These show that the CNN can effectively map the stress contourto the applied external strain. Such a map may play an important role inlinking microstructure information with macroscale information, e.g.,connecting microstructure failure strength to a macroscale strain state.This will assist the inverse design problem where the optimum loadingstate is inferred using local information.

Difference_(CNN) ^(s)=∥

_(CNN)(σ^(s)(α,β))−ε^(M,s(SCA))∥₂ .,s=1,2, . . . ,N _(V)  (1-33)

Convolutional Neural Network for Classification with Database Generatedby SCA

The application of a CNN to material microstructure predictions is notlimited to the sample problem shown above. Another example use of a CNNis as a microstructure classifier; this is similar to its commonapplication in image classification. By using a microstructure mesh andan applied strain on the microstructure as the input, a CNN can betrained to predict whether the microstructure will become damaged.

Training

The training procedure for the classification CNN is given by

$\begin{matrix}{\mspace{79mu}{{{{find}\text{:}\mspace{11mu} W_{mn}^{l}},{b_{n}^{l}\mspace{14mu}\left( {l = {2,3\mspace{11mu}\ldots\mspace{11mu} N_{L}}} \right)},{{in}\mspace{14mu}{FFNN}}}{W_{\xi,\psi}^{\zeta,\kappa,\eta},{b^{\kappa,\eta}\;\left( {\kappa = {1,2\mspace{11mu}\ldots\mspace{11mu} N_{kernel}}} \right)},\left( {\eta = {1,2\mspace{11mu}\ldots\; N_{comv}}} \right),{{in}\mspace{14mu}{CNN}}}\mspace{20mu}{\min\mspace{14mu}{loss}\mspace{14mu}{function}\text{:}}{{{cross}\mspace{14mu}{entropy}} = {\frac{1}{N_{T}}{\sum_{s = 1}^{N_{T}}\left( {- \left( {{d^{*s}{\log\left( d^{s} \right)}} + {\left( {1 - d^{*s}} \right){\log\left( {1 - d^{s}} \right)}}} \right)} \right)}}}{{{where}\text{:}\mspace{14mu} d^{s}} = {\mathcal{F}_{FFNN}\left( {\mathcal{F}_{flatten}\left( {\mathcal{F}_{pcp}^{conv}\left( {\ldots\mspace{11mu}{\mathcal{F}_{pcp}^{2}\left( {\mathcal{F}_{pcp}^{1}\left( {\sigma^{s}\left( {\alpha,\beta} \right)} \right)} \right)}\mspace{11mu}\ldots}\; \right)} \right)} \right)}}{{{nested}\mspace{14mu}{operation}\text{:}\mspace{11mu}{\mathcal{F}_{pcp}^{2}\left( {\sigma^{s}\left( {\alpha,\beta} \right)} \right)}} = {\mathcal{P}_{pooling}^{\eta}\left( {\mathcal{C}_{conv}^{\eta}\left( {\mathcal{P}_{padding}^{\eta}\left( {\sigma^{s}\left( {\alpha,\beta} \right)} \right)} \right)} \right)}}}} & \left( {1\text{-}34} \right)\end{matrix}$

The output of the classification CNN, d_(s), is a binary indicator: 0for non-damaged, 1 for damaged. Since the output is a binary value, theobjective function is now defined in terms of the cross entropy betweenthe truth value d^(*s) and the predicted value d^(s). Cross entropy, orlog loss, is widely used to measure the performance of a classificationmodel. The CNN in this section is trained on the data extracted from thedatabase used for the previous NNs. The database for training theclassifier includes pairs of micro stress distributions and damageindicators. In this case, a critical-stress-based damage criterion isused to decide whether an RVE is damaged or not: if any von Mises stressσ(X) exceeds a critical von Mises stress σ* the RVE is considereddamaged.

Prediction

Once trained, the CNN can predict whether the applied loading willresult in microstructure damage without carrying out the full RVEsimulation:

$\begin{matrix}{{\mathcal{F}_{CNN}^{classify}\left( {\sigma^{s}\left( {\alpha,\beta} \right)} \right)} = {d^{s} = \left( \begin{matrix}{0,} & {{Non}\text{-}{damaged}} \\{1,} & {Damaged}\end{matrix} \right.}} & \left( {1\text{-}35} \right)\end{matrix}$

Such a CNN will be used in a microstructure-based topology optimizationexample to illustrate the effect of a microstructural damage constrainton the optimized structure.

Microstructure-Based, Multiscale Topology Optimization Using NeuralNetworks

In this section, we will illustrate how FFNNs and CNNs might be used intopology optimization to achieve microstructure-based design. Thisdifferentiates the current approach from classical topology optimizationwhich typically uses simple constitutive relationships. As explainedabove, we propose to compress the RVE response database into an FFNN forforward prediction of RVE stress responses, where it will act similarlyto a traditional homogenized constitutive model. However, because adatabase of RVE responses is used, no functional form of thehomogenization is required. The RVE microstructure damage responses isrepresented with a trained FFNN+CNN; this introduces microstructuredamage, linked with the applied strain state of the RVE. By usingwell-trained FFNNs and CNNs, two different optimization problems aredefined as below:

1. Topology optimization with a material constitutive law extracted froman FFNN trained on the stress-strain relationship of a givenmicrostructure. In this case, a non-linear material behavior duringtopology optimization is used to achieve a design that is durable underextreme loading conditions where the material response enters thenon-linear region.

2. Topology optimization with constraints defined by the FFNN+CNNframework to identify microstructure damage and thereby design durable(damage aware) structures, using the CNN. In this case, themicrostructure damage acts as an extra constraint to the topologyoptimization formulation to achieve a design that alleviates or avoidspossible local microstructure damage.

In these two example problems, the design zone is described with a 60×30mesh of rectangular, linear elements. Each element is a 1 cm×1 cmsquare. The elastic material properties are from the homogenized SCAresults: E=200 MPa and v=0.27. An approximated non-linear elasticmaterial response is extracted from the RVE simulation obtained fromSect. 3, as described in the following sections. Note that whileplasticity is used, our FFNN is only valid for a non-linear elasticapproximation of the material response. A point load of 75 N is appliedat right bottom corner. The desired volume fraction of the optimizedpart is set as 0.35 of the original design zone. For the second casewith damage the critical stress is defined as 0.7 MPa.

Topology Optimization with FFNN

The formulation of microstructure sensitive topology optimization withan FFNN is shown in Eq. (1-36). The objective function is defined as theoverall strain energy of the structure, Φ, which is to be minimized. Asubtle but important difference in the present example in this work isthat the FFNN is used to replace the usual linear elastic materialresponse with a nonlinear one. By using a nonlinear material responsesdatabase depicted in above, a new avenue for data-driven material andstructure design is illustrated. This is depicted graphically in FIG.33, which shows the use of an FFNN to generate microstructure-basedstress-strain response within a topology design framework.

minimize ⁢ : ⁢ ⁢ Φ = ∫ Ω M ⁢ fu ⁢ ⁢ d ⁢ ⁢ Ω M + ∫ ∂ Ω M ⁢ tu ⁢ ⁢ dS M , ∀ u ∈ U * ⁢ ⁢with ⁢ : ⁢ ⁢ Φ ≡ ∫ Ω M ⁢ σ ⁡ ( X M ) ⁢ ɛ ⁡ ( X M ) ⁢ d ⁢ ⁢ Ω M , ∀ X M ∈ Ω M ⁢ ⁢subject ⁢ ⁢ to ⁢ : ⁢ ⁢ V ⁡ ( ρ ⁡ ( X M ) ) ≤ V * ⁢ ⁢ 0 ≤ ρ ⁡ ( X M ) ≤ 1 , ∀ X M ∈Ω M ⁢ ⁢ σ ⁡ ( X M ) = ρ ⁡ ( X M ) ⁢ FFNN ⁢ ( ɛ ⁡ ( X M ) ) , ⁢ ∀ X M ∈ Ω M ←FFNN ⁢ ⁢ for ⁢ ⁢ microstructural ⁢ ⁢ response , ⁢ where ⁢ : ⁢ ⁢ V ⁡ ( ρ ⁡ ( X M ) )= 1 Ω M ⁢ ∫ Ω M ⁢ ρ ⁡ ( X M ) ⁢ d ⁢ ⁢ X M ⁢ ⁢ Ω M ⁢ : ⁢ ⁢ macro ⁢ ⁢ domain , Ω ⁢ : ⁢ ⁢micro ⁢ ⁢ domain ⁢ ⁢ X M ⁢ : ⁢ ⁢ coordinate ⁢ ⁢ in ⁢ ⁢ macro ⁢ ⁢ domain , ⁢ X ⁢ : ⁢ ⁢coordinate ⁢ ⁢ in ⁢ ⁢ micro ⁢ ⁢ domain ( 1 ⁢ - ⁢ 36 )

where f is the body force, t is the applied traction on the boundary ofthe design zone, and u is the local displacement in the design zone. U*is the admissible displacement field, and S^(M) defines the boundariesof the macrostructure (the design region). σ(X^(M)) and σ(X^(M)) aremacro stress and strain. The density for each macro mesh is ρ(X^(M)).The desired volume of material remaining in the design zone is V* anddefined as 0.35, and V(ρ(X^(M))) is the optimized volume in the designzone. The density of location X^(M) is ρ(X^(M)) in the design zone, andthe homogenized stress is σ(X^(M)), defined as the product of ρ(X^(M))and

_(FFNN)(ε(X^(M)). Here,

_(FFNN) represents a trained FFNN that will generate stress predictionsbased on given strain input ε(X^(M)), as defined in Eq. (1-17)

In this case, the FFNN is used to approximate the RVE responses. Hence,σ^(M) is directly approximated by the FFNN. In truth, constraints of thesoftware used for optimization require a functional description of thebehavior (this limitation will be addressed in future work), thus theeffective von Mises RVE stress versus effective strain curve isapproximated using an exponential function: σ ^(M)=0.7784*e^(22.18*ε)^(M) −0.8071*e^(−378*ε) ^(M) . This hyperelastic material definition wasfit to the FFNN results. This ad-hoc approach ensures stability of theoptimization process by minimizing the overall strain energy usingcondition-based optimization.

FIG. 33 is topology optimization setup with FFNN. The FFNN is used tocompute non-linear material responses to drive for a new design. Thisreplaces the constitutive law commonly used for the macroscale with ahomogenized response of the microstructure for each point in themacroscale. Mathematically, σ^(M)(X^(M))=ρ(X^(M))

_(FFNN)(ε(X^(M))),∀X^(M)∈^(M), as defined in Eq. (1-36).

The optimized beam structures are illustrated in panels (a)-(b) of FIG.34 for linear elastic material and non-linear elastic material,respectively. Necessary results are provided in Table 1-7. Twostructures show substantial difference in the final shape of thestructure. This means the material non-linearity plays an important rolein topology optimization, where a new truss structure is realized inorder to ensure minimal strain energy. The result suggests theimportance of considerable of microstructure-based materialnon-linearity into structure optimization. It may also possible toinclude microstructure variation, such as different particle volumefractions, into the structure. In short, FFNN provides an alternativefor a data-driven microstructure-based topology optimization, where themicrostructural effect can be incorporated into the process and toachieve different designs that meet the design criteria.

TABLE 1-7 Results of the FFNN-based non-linear-elastic optimizationproblem. Note that substantial speedup is achieved while retaining theaccuracy and microstructural basis of the concurrent approaches. Thiswould provide further speed advantages in 3D. linear FFNN FE-SCA FE-FEmaterial (nonlinear) concurrent* concurrent* Initial compliance 12.6(6.3) 20.0 (10.0) — — (strain energy) (N cm) Optimized compliance 28.0(14.0) 38.0 (19.0) — — (strain energy) (N cm) Database generation + 0313 — — training (s) Optimization 338  472 23 328 220 × 10⁶ calculationtime (s) Factor of speed-up — 280 255     9 431 — over FE-FE No. ofiterations 14   18 — — *estimated, assuming the same number ofiterations as FFNN

Topology Optimization with Constraints Defined by FFNN+CNN

Topology optimization results may have high stress concentration zones.If not treated properly, the concentration zones may cause unexpecteddamage and affect the function of the structure. A traditional way toaddress this is to add stress or strain constraints to optimization.Previous authors have focused on optimization with stress concentrationand singularities. In these previous studies, most damage criteria areonly related to the macroscopic material model and do not consider microstructure mechanical behavior. In this work, an FFNN+CNN frameworktrained by the database generated with SCA provides amicrostructure-based prediction of damage for the damage criterion.

In order to do this, the FFNN defined in Eq. (1-25) that performs theoperation σ(X)=

_(FFNN) ^(micro)(ε^(M)) is used. As mentioned above, this predictslocal, rather than homogenized, stresses in the RVE. These local stressdistributions serve as input to the CNN, which indicates whether or notdamage has occurred. The optimization formulations are thus:

minimize ⁢ : ⁢ Φ = ⁢ ∫ Ω M ⁢ fu ⁢ ⁢ d ⁢ ⁢ Ω M + ∫ ∂ Ω M ⁢ tu ⁢ ⁢ dS M , ∀ u ∈ U * ≡⁢U T ⁢ KU = ∑ e = 1 N ⁢ ( ρ ⁡ ( X M ) ) p ⁢ u e T ⁢ k 0 ⁢ u e ⁢ ⁢ ⁢ subject ⁢ ⁢ to ⁢: ⁢ ⁢ V ⁡ ( ρ ⁡ ( X M ) ) ≤ V * ⁢ ⁢ ⁢ 0 ≤ ρ ⁡ ( X M ) ≤ 1 ⁢ ⁢ ∀ X M ∈ Ω M ⁢ ⁢ ⁢ ∀ X M∈ Ω M ⁡ ( X M ) , ∀ X ∈ Ω ⁡ ( X ) ⁢ : ⁢ ⁢ ⁢ CNN classify ⁢ ( σ ⁡ ( X ) ) = 0 ←Microscale ⁢ ⁢ damage ⁢ ⁢ criterion , ⁢ where ⁢ : ⁢ ⁢ σ ⁡ ( X ) = FFNN micro ⁢ ( ɛM ) ← Microscale ⁢ ⁢ stress ⁢ ⁢ prediction , ⁢ ⁢ Ω M ⁢ : ⁢ ⁢ macro ⁢ ⁢ domain , Ω ⁢: ⁢ ⁢ micro ⁢ ⁢ domain ⁢ ⁢ ⁢ X M ⁢ : ⁢ ⁢ coordinate ⁢ ⁢ in ⁢ ⁢ macro ⁢ ⁢ domain , ⁢ ⁢ X ⁢ :⁢⁢coordinate ⁢ ⁢ in ⁢ ⁢ micro ⁢ ⁢ domain ( 1 ⁢ - ⁢ 37 )

where Φ=Σ_(ε=1) ^(n)(ρ(X^(M)))^(p)u_(e) ^(T)k₀u_(e) is the compliance ofthe overall structure, N is the number of elements, p is thepenalization power (typically p=3), u_(e) is the displacement for eachelement, and k₀ is the element stiffness. The averaged strain responseof the RVE is ε^(M). The local stresses within the RVE are defined as σ,and are calculated with a trained FFNN

_(FFNN) ^(micro). Other variables are the same as defined above for theFFNN-only optimization. For any X in RVE Ω^(m)(X), the output value of

_(CNN) ^(classify) should be 0, which represents a non-damaged state.

Since the FFNN+CNN database only gives a criterion, sensitivity analysisis not preferred in this case. The algorithm here follows a refinedoptimally criteria (OC) method. The optimization program structure isbased on the 99-line topology optimization code (and thus the problem issimilar, though the implementation is not identical to the FFNN-basedoptimization above). During the density update, for each element, threestrain components will be passed to the FFNN+CNN. The FFNN+CNN willdetermine whether the current strain state is acceptable by assessingthe local microstructural response. This is shown schematically in FIG.35. For each material point within the design zone, the FFNN is used tocompute the material response, be it linear or non-linear, consideringthe effect of microstructure. The CNN is used to incorporatemicrostructure damage, which will drive the optimization algorithm for anew design compared to a topology optimization with only linearmaterial. Mathematically, this is ∀X^(M) ∈^(M) (X^(M)): σ(X)=

_(FFNN) ^(micro)(ε^(M)(X^(M))), ∀X∈(X): d(X^(M))=(

_(CNN) ^(classify)(σ(X))), as defined in Eq. (1-37). If any d(X) ismarked as damaged in the microstructure, the X^(M) point in the designzone containing that microstructure will be marked as damaged. If anelement has been damaged, the density of this element will be increasedby applying a penalty factor, while the densities of the rest of theelements will be decreased to satisfy the volume constraint.

Similarly to the FFNN example, panel (a) of FIG. 36 shows the referencecase: a linear elastic optimization without a damage constraint. Thefinal compliance is 30 N·cm. Panel (b) of FIG. 36 shows the optimizeddesign with a microstructure-based damage constraint. The finalcompliance is 31 N·cm. Notice that while the compliance is quitesimilar, the design under a microstructural damage constraint resemblesa more conventional truss structure; the optimization has avoided sharpangles and has fewer beams that give rise to stress concentrationslikely to result in microstructure-driven damage.

A summary of the design variables and important parameters related tothe optimization is provided in Table 1-8. The simple examples aboveshow the potential application of an FFNN+CNN database generated byclustering reduce order methods. However, the optimization is just basedon an artificially defined optimality criterion. The algorithm may notbe stable for all kinds of problems. In the future, we should study thesensitivity and singularity of constraints based on an FFNN+CNNdatabase.

TABLE 1-8 Results of the FFNN + CNN constraint optimization problem.Note that substantial speedup is achieved while retaining the accuracyand microstructural basis of the concurrent approaches. This wouldprovide further speed advantages in 3D. Linear FE-SCA FE-FE materialFFNN + CNN concurrent* concurrent* Initial compliance 295 (148) 295(148) — — (strain energy) (N · cm) Optimized compliance 30.0 (15.0) 31.0(15.5) — — (strain energy) (N · cm) Database generation + 0 512 trainingtime(s) Optimization 12.6 14.5 69 674 660 × 10⁶ calculation time (s)Factor of speed-up — 45 × 10⁶  9 473 — over FE-FE No. of iterations 5361 — — *estimated, assuming the same number of iterations as CNN

SUMMARY

Two challenges with current approaches to machine learning methods inthe mechanical science of materials are: (1) the database generationtime and effort are extensive, and (2) the application of machinelearning is not well developed or understood by the community. Thisstudy covers several different topics related to these challenges:

-   -   We have outlined, related, and compared three different        clustering-discretization methods (SCA, VCA, and FCA) that rely        on unsupervised learning for order reduction and the solution of        mechanistic governing equations for prediction.    -   One of these methods, SCA, was used to develop an example        material behavior database suitable for training neural        networks. This approach to database development substantially        reduces the effort required to acquire the information upon        which neural networks may be trained.    -   The basic operations, and how these combine to make predictions        of mechanical responses, in an FFNN were outlined. This includes        the role of weights, biases and activation functions as well as        the description of the training stage of the neural network as a        minimization problem using notation common within the mechanical        sciences.    -   A similar description of convolutional neural networks was        developed for two different possible applications: (1) to solve        inverse problems where the boundary conditions need to be        identified from a known stress distribution and (2) as a        classifier to identify if damage will occur within a        microstructure given a known stress distribution.    -   Two microstructure-sensitive topology optimizations are        demonstrated. In the first case, the material response at the        microscale derived from the FFNN results, and used to perform        design against a load that causes the material to behave in a        non-linear elastic way. In the second case, a material damage        constrain is added to the optimization, where the CNN is used to        identify if damage has occur on the microscale and penalize the        design accordingly.

In short, we have provided methods to more rapidly produce the dataneeded to train neural networks, developed further insight into theworking of neural networks from a mechanical sciences perspective, andhighlighted the potential for these methods to enhance practical designtasks. The database of responses made with SCA, codes used for trainingand prediction with the neural networks, and the topology optimizationcodes are developed. This will encourage the use of data science andmachine learning as a tool for mechanistic analysis, rather than simpleas an unknown black-box operator.

Several areas where further investigation might be useful have alreadybeen noted:

-   -   Further development of clustering methods to represent large        deformations, better capture anisotropic behavior or behavior        that changes due to loading conditions, and even refine clusters        during the prediction stage might be promising. The development        of contact or self-contact formulations applicable to clustering        discretization methods would aid in generality.    -   Formulations of clustering discretization solutions applicable        to the component scale (rather that only the RVE scale), and the        extension of concurrent multiscale solutions that use clustering        discretization at multiple scales are currently under        development.    -   For neural networks, methods to include history-dependence        (e.g., plasticity) are currently an active area. Including        physics in the neural network directly is another developing        area, e.g., with physics-informed neural networks (PINNs).    -   For optimization, sensitivity analysis for topology optimization        with FFNN and CNN should be further developed. More flexible        software to support this would also be desirable.    -   Multiscale topology optimization with various material        microstructure databases is still a developing area. The        approach outlined here may be a promising method to        simultaneously optimize topology and microstructure given        sufficient constraints.    -   The “data-driven” component of these methods (both        clustering-based discretization and neural networks) are not        restricted to the use of computational data. Information from        other sources, e.g., experimental sensor data and images, could        be included if it is available. If mixed data streams are used        extra care in data representation would be required.    -   Continuous validation and verification studies will help make        these methods robust and reliable.

Example 2 Data Science for Finite Strain Mechanical Science of DuctileMaterials

In this example, a mechanical science of materials, based on datascience, is formulated to predict process-structure-property-performancerelationships. Sampling techniques are used to build a trainingdatabase, which is then compressed using unsupervised learning methods,and finally used to generate predictions by means of mechanisticequations. The method presented in this example relies on an a priorideterministic sampling of the solution space, a K-means clusteringmethod, and a mechanistic Lippmann-Schwinger equation solved using aself-consistent scheme. This method is formulated in a finite strainsetting in order to model the large plastic strains that develop duringmetal forming processes. An efficient implementation of an inclusionfragmentation model is introduced in order to model this micromechanismin a clustered discretization. With the addition of a fatigue strengthprediction method also based on data science,process-structure-property-performance relationships can be predicted inthe case of cold-drawn NiTi tubes.

Increasing research efforts in fine scale experiments and numericalmodeling in recent decades have progressively led to a change inmodeling approaches in mechanics and materials science. Empirical andphenomenological material laws that were previously used to model thenonlinear mechanical response of structures and materials are beingreplaced by microstructure-based mechanistic material laws. Underarbitrary loading conditions the number of microstructure observationsand conditions to be modeled make the effort required for such anendeavor untenable for practical applications. The appeal of datascience and in particular machine learning is a drastic reduction in thenumber of microstructure observations and simulations required togenerate predictive material laws. There is hence a great interest in adata science theory for mechanical science of materials that couldgenerate predictive material laws from a predefined database ofexperimental and numerical results.

Multiple approaches have been proposed in the literature to reach thisgoal, generally summarized by three steps: (1) collecting data usinghigh-fidelity experiments and simulations to build a training database;(2) compressing the training database using unsupervised learningmethods for dimension reduction; and (3) generating predictions usingsupervised learning methods or mechanistic equations on the compressedtraining database and optionally cross-validating those predictionsusing a testing database with new high-fidelity experiments andsimulations.

The training database can be generated using, e.g., random sampling,Gaussian processes, or Sobol sequences. Because those sampling methodsmay require a lot of data points to cover the solution spacesufficiently for accurate predictions, deterministic sampling methodshave been considered by some authors. For instance, instead ofconsidering a large number of arbitrary, random loading conditions forthe training database, only 6 orthogonal loading conditions of smallamplitude were proved to be sufficient for small strain elastoplasticanalysis.

Compression of the training database can be achieved using variousunsupervised learning methods for dimension reduction, such as ProperOrthogonal Decomposition (POD), K-means clustering and self-organizingmaps. The choice of compression method has a significant importance asit defines the discretization of mechanistic equations that will besolved in the prediction stage. POD leads to shape functions of globalsupport, while clustering methods ensure a cluster-wise discretization.

As a result of data compression, the complexity of high-fidelityexperiments and simulations that were used to build the trainingdatabase is encapsulated in a few degrees of freedom. In order to solvefor those degrees of freedom and predict mechanical response atarbitrary loading conditions, mechanistic equations have to bereformulated in terms of the reduced degrees of freedom. This newformulation of mechanistic equations is usually called a reduced ordermodel, although this denomination encompasses approaches such as propergeneralized decomposition which do not rely on data science.

Additionally, some approaches couple the data compression andmechanistic prediction steps to improve the reduced order model duringthe simulation. Some supervised learning methods have been applieddirectly to the training database with a built-in compression stage.This is the case for instance for artificial neural networks, which havebeen applied in the literature to predict mechanical properties ofmaterials as a function of their microstructural characteristics.

In this exemplary study, we disclose a data science mechanistic approachfor ductile materials by Self-consistent Clustering Analysis (SCA), adata-driven mechanistic material modeling theory developed for smallstrain elastoplastic materials. SCA relies on data compression throughclustering and mechanistic prediction through micromechanics andhomogenization theory.

In this study, the mechanistic equations that SCA relies on to makepredictions are reformulated for finite strain elastoplastic materials.Numerical convergence of this new method is verified. This newformulation of SCA enables the prediction of the nucleation of voids inductile materials by debonding and fragmentation of inclusions at thescale of their microstructure, which is shown in FIG. 37, where ductilematerials' microstructures are discretized using voxel meshes withmatrix shown in blue and inclusions in red: panel (a) two-dimensionalmicrostructure, and panel (b) inside view of a three-dimensionalmicrostructure with a fragmented inclusion surrounded by a debondingvoid shown in light gray. This prediction is achieved with a complexityreduced by several orders. This advantage is exploited to predictprocess-structure-property relations for cold drawn Nickel-Titanium(NiTi) tubes.

Data Science Formulation

Microstructure-based material modeling requires the definition of anidealistic or statistically representative microstructure realization,called RVE. Homogenized material laws can be computed by analytically ornumerically solving a boundary value problem for the response of thatRVE. For arbitrary microstructure geometries and complex behavior ofmicrostructure constituents (plasticity, fracture), numerical methodssuch as the Finite Element (FE) method or Fast Fourier Transform(FFT)-based numerical methods are required.

The microstructures that will be studied in the present paper correspondto ductile materials and feature one or multiple inclusions and voidsembedded in a matrix, as shown in FIG. 37. The complexity of themicrostructure's constituents' behavior arises due to thehyperelastoplastic response of the matrix, the hyperelastic-brittlebehavior of the inclusions, and debonding micromechanisms at thematrix/inclusions interface.

The FE method can be used with any structured or unstructured FE mesh ofthe undeformed RVE domain Ω₀ ^(m) (the superscript m means microscopic),while FFT-based methods require structured voxel meshes such as thatshown in FIG. 37. In the FE method discrete equations are written forthe displacement field u^(m), which is approximated at mesh nodes as

u ^(m)(X)≈Σ_(n=1) ^(N) ^(nodes) u ^(m,n) N ^(n)(X),X∈Ω ₀ ^(m),  (2-1)

where N_(nodes) is the number of nodes in the FE mesh, u^(m,n) is thedisplacement vector at node n, and N^(n) is the FE shape function atnode n. In FFT-based numerical methods, discrete equations are writtenfor the deformation gradient tensor field F^(m)=I+∇_(X)u^(m), which isapproximated voxel-wise as

F ^(m)(X)≈Σ_(n=1) ^(N) ^(voxels) F ^(m,n)χ^(n)(X),X∈Ω ₀ ^(m),  (2-2)

where N_(voxels) is the number of voxels, F^(m,n) is the deformationgradient tensor in voxel n, and χ^(n)(X) is the characteristic functionwhich is equal to 1 if X is inside voxel n and zero otherwise.

For a given microstructure, the displacement field u^(m) and thedeformation gradient field F^(m) depend on boundary conditions appliedto the RVE. In the present work, u^(m) will be decomposed over the RVEdomain Ω₀ ^(m) into a linear part and a periodic part. As a result,F^(m) will be decomposed into a constant part F^(M) (the superscript Mmeans Macroscopic) and a periodic part with zero average over Ω₀ ^(m).These assumptions correspond to first order homogenization theory.

Data science is used in the mechanical science of materials to predicteither u^(m) or F^(m) as a function of F^(M). As stated in theintroduction, the first step is to generate data through simulations.Simulation results in the training database will have large dimensionsdue to dependence of approximations in Eqs. (2-1) and (2-2) on eitherthe number of nodes or the number of voxels. Data compression isnecessary to obtain new approximations with reduced dimensions.

Data Compression

Dimension reduction can be achieved using various methods among whichPOD and clustering are presented and compared in the following.

The general formulation of data science approaches that is developedherein is only relevant if the complexity of simulations that are to beconducted in the prediction stage is at least one order superior to thecomplexity of simulations required in the training and data compressionstage. The relevance of data science approaches also depends on theamount of work that can be transferred out of the prediction stage. Thiswill be evidenced in the following in the case of POD and clusteringbased data science approaches for mechanical science of materials.

Data compression in the case of POD consists in replacing the largenumber of local FE shape functions (N^(n))_(n=1 N) _(nodes) byK<<N_(nodes) global functions (W_(i) ^(k))_(k=1 K,i=13), calledprincipal components or modes. The latter can be computed using variousdecomposition techniques such as principal component analysis orsingular value decomposition. The resulting approximation replacing Eq.(2-1) is

u _(i) ^(m)(X)≈Σ_(k=1) ^(K) u _(i) ^(m,k) W _(i) ^(k)(X),X∈Ω ₀,  (2-3)

where the modes are discretized at mesh nodes as

W _(i) ^(k)(X)=Σ_(n=1) ^(N) ^(nodes) W _(i) ^(k,n) N ^(n)(X),X∈Ω₀.  (2-4)

Simulations in the prediction stage can then be conducted using astandard FE weak form but replacing approximation of Eqs. (2-1) by(2-3). It is interesting to see that once the modes are computed in thedata compression stage, Eq. (2-4) can be precomputed at integrationpoints of the FE mesh in the same way that FE shape functions areusually precomputed in FE codes.

However, if the material is heterogeneous, or if it has a nonlinearbehavior that leads to heterogeneous deformations, material integrationstill has to be solved at each integration point. Consequently, in thePOD method, the complexity of material integration is not reduced.Additionally, the stiffness matrix associated to the FE weak form isdense because of the form of Eq. (2-4), and hence its solution usingdirect or iterative solvers has a cubic worst-case complexity instead ofquadratic. However, this complexity depends on K instead of N_(nodes),with K>>N_(nodes), and is therefore drastically reduced by POD.

Data compression in the case of SCA follows a different approach, wherethe initial numerical method is FFT-based. The large number of voxels isto be replaced by K<<N_(voxels) mutually-exclusive groups of voxels thatare called clusters and that span the entire RVE domain. Clusters can beconstructed using various clustering techniques such as K-meansclustering, or self-organizing maps. Examples of data that can be usedfor clustering are given below. The resulting approximation replacingEq. (2-2) is

F ^(m)(X)≈Σ_(k=1) ^(K) F ^(m,k)χ^(k)(X),X∈Ω ₀ ^(m),  (2-5)

where F^(m,k) is the cluster-wise constant deformation gradient tensorin cluster k, and χ^(k)(X) is the characteristic function which is equalto 1 if X is inside any voxel of cluster k, and zero otherwise. Becausein the FFT-based numerical method the degrees of freedom are directlythe voxel-wise constant deformation gradients, interpolation andintegration are carried out at the same points. Thus, clustering degreesof freedom directly leads to a reduction of the number of degrees offreedom and of material integration complexity. In fact, in SCA, thecomplexity of all operations conducted in the prediction stage onlydepends on the number of clusters K, with the most expensive operationbeing, similarly to POD, the solution of a dense linear system. Thelatter results from the reformulation and discretization of the Cauchyequation into the discrete Lippmann-Schwinger equation. These steps aredescribed in the following in the finite strain case following recentwork on finite strain FFT-based numerical methods and then integratingit into SCA.

Continuous Lippmann-Schwinger Equation

As mentioned previously, first order homogenization consists in definingthe deformation gradient tensor field in the RVE F^(m) as the additionof the macroscopic (homogeneous) deformation gradient F^(M) and amicroscopic (heterogeneous) fluctuation. Hill's lemma can be used todefine the macroscopic first Piola-Kirchhoff stress tensor P^(M) as theaverage of the microscopic one

$P^{M} = {\frac{1}{\Omega_{0}^{m}}{\int_{\Omega_{0}^{m}}{{P^{m}(X)}{{dX}.}}}}$

Hill's lemma requires (F^(m)−F^(M)) to verify compatibility, i.e., toderive from a periodic displacement field, and F^(m) to verifyequilibrium, i.e., to be the solution of the Cauchy equation

∇_(X) ·P ^(m)(F ^(m)(X))=0,X∈Ω ₀ ^(m).  (2-6)

It can be shown that Eq. (2-6) is equivalent to the Lippmann-Schwingerequation

$\begin{matrix}{{{F^{m}(X)} = {{- {\int_{\Omega_{0}^{m}}{{{\mathbb{G}}^{0}\left( {X,X^{\prime}} \right)}\text{:}\mspace{11mu}\left( {{P^{m}\left( {F^{m}\left( X^{\prime} \right)} \right)} - {{\mathbb{C}}^{0}\text{:}\mspace{11mu}{F^{m}\left( X^{\prime} \right)}}} \right){dX}^{\prime}}}} + F^{0}}},{X \in {\Omega_{0}^{m}.}}} & \left( {2\text{-}7} \right)\end{matrix}$

The fourth rank tensor

⁰ is the stiffness tensor associated to an isotropic linear elasticreference material. The far field deformation gradient tensor F⁰ and theperiodic Green's operator

⁰ are determined below. The latter maps any tensor field τ^(m) to acompatible one:

∃u∈(H ¹(Ω₀ ^(m)))³ ,u periodic on Ω₀ ^(m),−

⁰*τ^(m)=∇_(x) u,  (2-8)

where H¹(Ω₀ ^(m)) is the Sobolev space of square-integrable functionswhose weak derivatives are also square-integrable.

The combination of Eqs. (2-7) and (2-8) yields a microscopic deformationgradient tensor F^(m) that verifies compatibility and a firstPiola-Kirchhoff stress tensor P^(m) that verifies equilibrium.

Discrete Lippmann-Schwinger Equation

SCA includes solving Eq. (2-7) cluster-wise instead of voxel-wise. Thischoice is inspired from micromechanics and in particular TransformationField Analysis. FIG. 38 shows an example of clustering performed on themicrostructures in FIG. 37. In FIG. 38, panel (a) shows two-dimensionalmicrostructure discretized using 8 clusters; panel (c) shows sametwo-dimensional microstructure discretized using 65 clusters; and panel(c) shows three-dimensional microstructure discretized using 217clusters showing two clusters in the matrix phase (two shades of blue),one cluster in the inclusion phase (red), and one cluster in the voidphase (light gray).

As a result of the training stage, the RVE domain Ω₀ ^(m) is discretizedinto K subsets (Ω₀ ^(m,k))_(k=1 K). The degrees of freedom in theFFT-based numerical method are associated with the microscopicdeformation gradient F^(m). In SCA, F^(m) is discretized by acluster-wise constant approximation (F^(m,k))_(k=1 K). As a consequence,the microscopic first Piola-Kirchhoff stress tensor is also approximatedcluster-wise (P^(m,k))_(k=1 K), and Eq. (2-7) can be discretized as

F ^(m,k)=−Σ_(k′=1 K)

^(0,k,k′):(P ^(m,k′)−

⁰ :F ^(m,k′))+F ⁰ ,k=1K  (2-9)

where

⁰ is the interaction tensor defined by

$\begin{matrix}\begin{matrix}{{\mathbb{D}}^{0,k,{k\;\prime}} = {\frac{1}{\Omega_{0}^{m,k}}{\int_{\Omega_{0}^{m}}{{\chi^{k}(X)}{\int_{\Omega_{0}^{m}}{{\chi^{k\;\prime}\left( X^{\prime} \right)}{{\mathbb{G}}^{0}\left( {X,X^{\prime}} \right)}{dX}^{\prime}{dX}}}}}}} \\{= {\frac{1}{\Omega_{0}^{m,k}}{\int_{\Omega_{0}^{m,k}}{\left( {\chi^{k\;\prime}*{\mathbb{G}}^{0}} \right)(X){{dX}.}}}}}\end{matrix} & \left( {2\text{-}10} \right)\end{matrix}$

The characteristic functions χ^(k) and χ^(k′) are equal to 1 in,respectively, clusters k and k′, and 0 elsewhere. In the FFT-basednumerical method, the periodic Green's operator

⁰ depends on

⁰, and is known in closed form in Fourier space. Because

⁰ is related to an isotropic linear elastic reference material,

⁰ can be expressed in Fourier space as a function of the reference Laméparameters λ⁰ and μ⁰. It is then obtained in real space by using theinverse FFT. In particular, Eq. (2-10) can be written in the form

$\begin{matrix}{{{\mathbb{D}}^{0,k,{k\;\prime}} = {{{f^{1}\left( {\lambda^{0},\mu^{0}} \right)}{\mathbb{D}}^{1,k,{k\;\prime}}} + {{f^{2}\left( {\lambda^{0},\mu^{0}} \right)}\mspace{11mu}{\mathbb{D}}^{2,k,{k\;\prime}}}}},{{\mathbb{D}}^{i,k,{k\;\prime}} = {\frac{1}{\Omega_{0}^{m,k}}{\int_{\Omega_{0}^{m,k}}{{FFT}^{- 1}\left\{ {{FFT}\left\{ \chi^{k\;\prime} \right\}{\hat{\mathbb{G}}}^{i}} \right\}(X){dX}}}}},{i = {1,2.}}} & \left( {2\text{-}11} \right)\end{matrix}$

The detailed expressions of f¹, f²,

¹ and

² can be found among others. Drastic computational cost reduction isenabled by SCA thanks to a reduced number of degrees of freedom byclustering, and by the fact that

¹ and

² can be precomputed in the training stage. Therefore, neither FFTs norinverse FFTs are computed in the prediction stage, even if the referencematerial is changing.

In the present work, mixed boundary conditions are coupled to Eq. (2-9).Some components F_(i,j) ^(m) of the average of the microscopicdeformation gradient are set equal to their macroscopic counterpartsfrom F_(i,j) ^(M), and some other components P_(i,j) ^(m) of the averageof the microscopic first Piola-Kirchhoff stress tensor are set to zero.This can be done by adding the following conditions to Eq. (2-9):

$\begin{matrix}\left\{ \begin{matrix}{{{\sum_{k = {1\mspace{14mu} k}}{{\Omega_{0}^{m,k}}F_{i,j}^{m,k}}}\  = \ {{\Omega_{0}^{m}}F_{i,j}^{M}}},} & {\left( {i,j} \right) \in \mathcal{F}} \\{{{\sum_{k = {1\mspace{14mu} k}}{{\Omega_{0}^{m,k}}P_{i,j}^{m,k}}} = 0},} & {\left( {i,j} \right) \in {\left\{ {1,2,3} \right\}^{2} \smallsetminus \mathcal{F}}}\end{matrix} \right. & \left( {2\text{-}12} \right)\end{matrix}$

where

⊂{1,2,3}² is the set of components for which kinematic conditions areimposed.

As noted, solutions of Eq. (2-9) depend on the choice of referencematerial. An optimal choice can be computed in the prediction stage bymaking the reference material consistent with the homogenized material.This means that the far field deformation gradient tensor F⁰ is anadditional unknown that must be solved for in SCA, as opposed to theFFT-based numerical method where F⁰≡F^(M). The self-consistent methodconsists in using a fixed-point iterative method where, at each step,the reference Lamé parameters λ⁰ and μ⁰ are changed so that ∥P^(M)−

⁰: (F⁰−I)∥₂ is minimized. This is presented in algorithm form in theAppendix.

To summarize, SCA is based on a voxel-wise discretization of the RVEdomain, which is inherited from FFT-based numerical methods. Theoriginality of SCA comes from the use of a K-means clustering algorithmin the training stage to cluster voxels based on a mechanistic a prioriclustering criterion computed using a simple sampling of the loadingspace. This training stage also includes computing all voxel-wise andcomputationally expensive operations such as FFTs and inverse FFTs.

In the prediction stage, a self-consistent iterative algorithm is usedto search for the optimal choice of reference Lamé parameters. At eachiteration of this self-consistent loop, matrix assembly operations areaccelerated because all voxel-wise operations have been precomputed inthe training stage and already reduced to cluster-wise contributions. ANewton-Raphson iterative algorithm must be embedded within eachself-consistent iteration as we are considering nonlinear materials,thus the discrete Lippmann-Schwinger equation is linearized.

The outputs from SCA are the microscopic variables' cluster-wiseapproximations, including the microscopic first Piola-Kirchhoff stresstensor. The latter can be used to predict void nucleationmicromechanisms through stress-based fracture criteria as will bedescribed in Sec. 4.

The main advantage of SCA over POD techniques is that in the predictionstage all operations are conducted cluster-wise in SCA instead ofvoxel-wise, including material integration and even fragmentationmodeling.

Numerical Validation

Before considering a specific application, general ductile materialsmicrostructure are modeled in this section both using a finite strainFFT-based numerical method and finite strain SCA. The goal is tovalidate the numerical convergence of SCA towards the reference result,and its capability to compute accurate predictions with a reducedcomplexity, as was shown for the small strain case.

In the present large strain case, hyperelastic behavior is modeled bothin the matrix and in the inclusions. A multiplicative von Misesplasticity model with linear isotropic hardening is added to model thenonlinear response of the matrix. Inclusions are assumed to be brittle,which is a common assumption for hard phases in ductile materials. Themodel ductile material microstructure is shown in panel (a) of FIG. 39,and material properties are given in Table 2-1 for the matrix and theinclusions. The microstructure is discretized using 100×100×100 voxels,which is sufficient to accurately predict the response with the FFTnumerical method based on our preliminary calculations (not reportedherein).

TABLE 2-1 Material parameters for the model ductile material ParameterMatrix Inclusion Void Units Young's 70.0 400.0 70.0e−3 GPa ModulusPoisson's Ratio 0.33  0.2 0.0 — Yield Strength 400 — — MPa Hardening1333 — — MPa Modulus

The reference result is computed on the voxel mesh using the finitestrain FFT based numerical method under unidirectional tension up to 25%applied logarithmic strain with strain increments of size 0.001. Thetraining database used for K-means clustering simply includes thevoxel-wise deformation gradient tensor extracted from the firstincrement of this reference simulation, which corresponds to a linearelastic analysis. In other words, the simulation used for training isidentical in all aspects (mesh, geometry, material properties) to theone conducted in the prediction stage, except for loading conditionsbecause only one strain increment is applied during training. Thisdatabase can hence be constructed with a negligible computational cost.The K-means algorithm requires a predefined number of clusters, which isvaried from k₁=1,4,16,64,256 in the matrix phase, and k₂=1,1,4,13,26 inthe inclusion phase.

The comparison between the reference macroscopic stress/strain curve andthose predicted by finite strain SCA is presented in panels (b)-(c) ofFIG. 39. For 16 clusters in the matrix and more, a very accurateprediction is obtained with finite strain SCA. This extends thevalidation conducted to large strains.

To show the influence of clustering on computational complexity, acomparison of computation times is presented in panel (c) of FIG. 39.The computation time is shown to be reduced by four orders using SCAcompared to the FFT based numerical method. This shows that SCAdrastically reduces the complexity of microstructure calculations, basedon a mechanistic clustering of voxels.

This interesting advantage of SCA is demonstrated in a second set ofsimulations where the inclusions in panel (a) of FIG. 39 are replaced byvoids. Material properties for this porous ductile material are the sameas those in Table 1, except that the stiffness of the voids is assumedto be 1% of that of the matrix. Loading is set to uniaxial tension toensure that the stress state remains constant during the analysis.

The comparison between the reference macroscopic stress/strain curve andthose predicted by finite strain SCA is presented in panels (a)-(b) ofFIG. 40. It can be seen that convergence is much slower for this examplewith voids, which features larger plastic strains than in the inclusionscase. However, predictions are very close to the reference result.

The main advantage of SCA in terms of computation time is preserved, asshown in panel (b) of FIG. 40. As all SCA predictions are very close,one could use a very small number of clusters and obtain a goodapproximation of the reference result, resulting once again in a drasticreduction of computational complexity.

These first simulations with the proposed finite strain SCA formulationare promising as predictions are close to the reference results for adrastically reduced computational complexity. However, this comparisonis purely global: only averaged results are being assessed. At largeplastic strain or for more complex loading conditions, veryheterogeneous and localized strain fields may develop and would requireimprovements to the method. For instance, adaptive clustering techniquescould be considered to update the clustering when plastic localizationphenomena occur within the RVE.

Fatigue Strength Prediction of Cold Drawn NiTi Tubes

The objective of this section is to demonstrate the utility of amechanical science of materials based on data science to predictprocess-structure-property-performance relationships. The chosenapplication is the prediction of the fatigue strength of cold drawn NiTitubes as a function of drawing ratio and initial inclusion Aspect Ratio(AR).

NiTi tubes are formed using a series of hot and cold metal formingprocesses coupled to heat treatments. They are used in the making of,e.g., arterial stents and heart valve frames that undergo a large numberof cyclic loads due to heart beats. A critical measure of a tube'sperformance is hence the fatigue strength of its material, which isitself a function of the material's fatigue life under different appliedcyclic strain amplitudes. This fatigue life can be predicted usingmicromechanical simulations, which depend on the microstructuralconstitution of NiTi tubes. The latter is a consequence of formingprocesses, and in particular of the cold drawing step, which is thefocus of the following study.

We disclose simulating cold drawing at the microscale by applyingbiaxial compression to an initially debonded inclusion embedded within aNiTi matrix. In order to predict the evolution of this microstructureduring cold drawing, the finite strain SCA theory introduced above iscompleted with a fragmentation model. The microstructure is extracted atdifferent stages of this drawing model and used as input to adata-driven fatigue life prediction model developed in previous studiesand described briefly. The transfer of the microstructure morphologyfrom the drawing model to the fatigue life prediction model requires adisplacement reconstruction and microstructure interpolation step thatis described. Process-structure-property-performance predictionsobtained using this data science approach are presented.

Cold Drawing Model

Using the self-consistent scheme presented above, the discreteLippmann-Schwinger equation (2-9) can be solved with mixed boundaryconditions (2-12) and appropriate constitutive models for themicrostructure's constituents. Constitutive models and materialparameters are kept identical to those used and reported in Table 2-1,but they are completed by an inclusion fragmentation model. In addition,inclusions are assumed to be initially debonded as a result of highshear stresses developing early at the inclusion/matrix interface duringcold drawing, similarly to the cold extrusion process.

Inclusions fragmentation is modeled using a Tresca yield criterionaveraged inclusion-wise, following a regularization technique used in aprevious work and coupled to a size-effect criterion. The Trescacriterion defines the shear stress σ_(Tresca) as

σ_(Tresca)=1/2 max(|σ₁−σ₂|,|σ₂−σ₃|,|σ₃−σ₁)  (2-13)

where σ₁, σ₂, σ₃ are the cluster-wise constant principal stressescomputed within each cluster of the inclusion phase. The shear stressσ_(Tresca) is then averaged inclusion-wise, or inclusion fragment-wiseif the inclusion has already broken-up, and that averaged value σ_(Tresca) is compared to the inclusion shear strength σ_(Tresca) ^(c).The equivalent radius of this inclusion or inclusion fragment r is alsocompared to a critical size parameter r^(c). If the shear strength hasbeen reached and r≥r_(c), the inclusion cluster containing theσ_(Tresca)-weighted barycenter of that inclusion or inclusion fragmentis turned into void, as illustrated in FIG. 41. In practice, thisconsists in reducing its Young modulus to a 1000th of its initial valueover several load increments. This procedure is carried out at the endof each load increment of the finite strain SCA simulation, once Eqs.(2-9) and (2-12) have been solved using the self-consistent scheme. Asrevealed in FIG. 41, the orientation of the fragmentation crack ispredefined and is initially orthogonal to the drawing direction. Thefracture criterion parameters are given in Table 2-2.

TABLE 2-2 Fracture criterion parameters for the cold drawing modelParameter Value Units Shear Strength 3000 MPa Critical 0.17 1/RVE sizeinclusion size

Fatigue Life Prediction Model

High-cycle fatigue life can be estimated based on local plasticdeformation predicted under cyclic loading, where these deformations arecomputed using the SCA outlined above. Since cyclic strain amplitudesare low, typically below 1% reversed strain, a small strain formulationcan be used. An appropriate micro-scale material law-crystal plasticity(CP)—is solved cluster-wise to obtain the cyclic change in plastic shearstrain (Δγ_(p)) and stress normal (σ_(n)) to that strain in the matrixmaterial. The peak value of these variables reach cyclic steady-staterapidly, typically within 3 or 4 loading cycles. From these, a scalarvalue called the Fatigue Indicating Parameter (FIP) can be defined,which quantifies the fatigue driving force at any location. Here, weadopt the Fatemi-Socie FIP, defined by

$\begin{matrix}{{FIP}{= {\frac{\Delta\gamma_{p}}{2}\left( {1 + {\kappa\frac{\sigma_{\mathfrak{n}}}{\sigma_{y}}}} \right)}}} & \left( {2\text{-}14} \right)\end{matrix}$

This FIP is a critical plane approach based upon the plane of maximumnormal stress, σ_(n), normalized by the material yield stress (σ_(y))and a material dependent parameter κ, which controls the influence ofnormal stress (here κ will be taken as 0.55). When used with SCA and aCP law, the FIP in each cluster is computed cluster-wise from plasticstrain and stress across time increments using a simple search tomaximize the plastic strain, and thus identify the critical plane. Themaximum, saturated FIP (NFIP_(max)) can be correlated to the number offatigue incubation cycles (N_(ine)) a microstructure can withstand usinga Coffin-Manson parameterization. By computing N_(inc) for a number ofdifferent strain amplitudes to generate a strain-life curve, the fatiguestrength (strain amplitude at which a given number of cycles can bereached) can be computed through interpolation or fitting.

The CP material law, calibrated to capture the hardening response of theB2 phase of NiTi, is used. Post-yield hardening is computed with abackstress term that accounts for direct and dynamic hardening.Consistent with the worst-case or nearly worst-case approximation forthe material phase, and the lack of crystallographic information fromthe cold drawing model, the matrix is assumed to be composed of a singlegrain oriented such that its Schmid factor is maximized. The hardinclusion phase (representing oxides or carbides) is taken to belinear-elastic with an elastic modulus 10 times that of the matrixmaterial. This process follows the CPSCA formulation integrates the CPmaterial law within SCA, and the fatigue prediction method shown forsynthetic microstructures. Indeed, the same model parameters are usedfor CP, FIP and Coffin-Manson as are shown.

Transfer of Microstructure from Cold Drawing Model to Fatigue LifePrediction Model

The fatigue life prediction model relies on CPSCA and hence on anunderlying voxel mesh. In order to use this model, the deformed andfragmented microstructures from the drawing simulation results need tobe transferred to a new voxel mesh.

The first step is to reconstruct the microscopic displacement vectorfield, since both FFT-based numerical methods and SCA only solve for themicroscopic deformation gradient tensor. This is done using a simpleTaylor expansion, or forward finite difference formula, which computesthe displacements at all nodes of the voxel mesh from the microscopicdeformation gradients inside voxels. This computation starts at somearbitrary corner of the RVE domain where the displacement is fixed tozero, which is in agreement with FE-based linear homogenizationimplementations.

Once the displacement vector field has been reconstructed, the mesh canbe deformed by adding these displacements to node coordinates, as shownin panels (a)-(b) of FIG. 42. Finally, the phase tags (matrix,inclusion, void) are transferred from this deformed mesh to a new voxelmesh of the deformed RVE domain using a simple voxel-wise constantinterpolation. As a result, the new voxel mesh is compatible withFFT-based numerical methods and SCA, but embeds phase tags thatcorrespond to a cold drawn microstructure. Fatigue life predictions canbe computed on this new voxel mesh using the CPSCA method, as shown inpanel (c) of FIG. 42.

Fatigue Strength Predictions

Three different initial conditions for the drawing model wereconsidered, representing possible variability in the quality and degreeof processing in the feed stock used for cold drawing. Each conditionincludes a single, ellipsoidal, debonded inclusion centered in thematrix with a different AR in the load direction. The cross-sectionalarea (i.e., the minor axis of the ellipsoid) of the initialconfiguration is kept constant, and the length (in the drawingdirection) changed. By doing this, we study the influence of AR (meancurvature) on drawing, fragmentation and subsequent fatigue life. Thesethree different cases are deformed to up to 60% section height reductionby applying biaxial compression with a stress-free third axis. For eachcase, at every 5% height reduction the procedure outline is performedand the fatigue strength is computed. In order to compute the fatiguestrength, fatigue lives at strain amplitudes ranging between 0.36% and0.54% in increments of 0.06% were computed, and the strain amplituderequired to reach 10⁷ cycles was estimated using a power-law fit to thatdata. Triangle waveforms with loading rate 0.1/s were used throughout toapply fully reversed tension-compression (strain ratio, R=−1) fatigueloading.

The results from this parametric study are shown in FIG. 43. In thecenter of the figure, fatigue strength—the maximum strain amplitude atwhich a specified number of cycles can be obtained—is plotted versus thesection reduction resulting from drawing. At each point in sectionreduction and for each AR the fatigue strength at 10⁷ cycles is computedusing the procedure outlined above. Cross-sections of the volume aregiven at six different points in order to visualize and analyze theprocess by which the fatigue strength changes with reduction percent.The drawing model captures the overall trend of an increase in fatiguelife with increasing height reduction, particularly for the highest AR.A large increase in the fatigue life is noted when the AR 3 particlefragments, between 40% and 45% reduction. The pre-fragmentationconfiguration is shown in the upper right-hand subset to FIGS. 2-7, andthe post-fragmentation configuration is shown in panel (c) of FIG. 42.The distribution of FIP changes noticeably between these two states,with the field much more concentrated at the interfaces beforefragmentation and more distributed throughout the volume afterfragmentation. Up to 60% height reduction neither AR 1 nor 2 fragment,and no substantial decrease in life is noted. This is consistent withexperimental experience, where relatively large reductions are oftenrequired to achieve large gains in fatigue performance. Higher reductionpercents or different material properties would be required to fragmentthese cases.

According to the embodiment, a general formulation of data scienceapproaches for mechanical science of materials is presented. Thisgeneral formulation includes reducing the complexity ofprocess-structure-property-performance prediction methods usingunsupervised learning methods on a training database of high-fidelitysimulations. This is evidenced in the case of a training databasecomposed of RVE simulations results computed under various loadingconditions using the FE method or FFT-based numerical methods. Dimensionreduction leads to a compressed RVE model where nodes and voxels arereplaced by either modes or clusters depending on the supervisedlearning method used for data compression.

In the prediction stage, supervised learning methods or mechanisticequations are solved using the compressed RVE. For instance, POD can beused to solve the Cauchy equation using a compressed FE discretizationwhere the complexity depends on the number of modes instead of thenumber of nodes. Similarly, K-means clustering can be used to solve theLippmann-Schwinger equation with a complexity depending on the number ofclusters instead of the number of voxels. The interesting advantage ofthis later approach over the former is that it reduces both thecomplexity of mechanistic equations solution and material integration.

The solution of the Lippmann-Schwinger equation using a clustereddiscretization requires a self-consistent scheme that has been extendedto finite strain elastoplastic materials and coupled to micromechanicalvoid nucleation models in this example. As a result, microstructureevolutions due to large plastic strains have been captured during thecold drawing of NiTi tubes. These microstructure evolutions have beenrelated to the fatigue life and then the fatigue strength of NiTi tubesusing a second data science based approach developed in a previous work.Therefore, it has been demonstrated that the proposed data scienceformulation for mechanical science of materials can predictprocess-structure-property-performance with a reduced complexity.

Example 3 Predictive Multiscale Modeling for Unidirectional Carbon FiberReinforced Polymers

This exemplary study presents a predictive multiscale modeling schemefor Unidirectional (UD) Carbon Fiber Reinforced Polymers (CFRP). Abottom-up modeling procedure is discussed for predicting the performanceof UD structures. UD material responses are computed from high fidelityRepresentative Volume Elements (RVEs). A data-driven Reduced OrderModeling (ROM) approach compresses RVEs into Reduced Order Models somaterial responses can be computed in a concurrent fashion along withthe structural level simulation. The approach presented in this exampleis validated against experimental data and is believed to provide designguidance for future fiber reinforced polymers development process.

In modern engineering applications, composite materials are receivinggrowing attention for their extraordinary lightweight and strength. Tounderstand the mechanical performance of various CFRP designs, physicaltests are necessary. With computational power continually growing, it isnow possible to utilize the Integrated Computational MaterialEngineering (ICME) approach to virtually evaluate the performance ofcomposite designs and provide design guidance for composite materials.The ICME approach directly integrates microstructure information intoproperty and performance prediction. In the ICME process, the intrinsicrelationship between material microstructure and mechanical performancecan be captured by a multiscale model which links microstructure tomacroscale performance. In this exemplary study, a bottom-up ICMEmodeling framework for UD CFRP is introduced. The framework incorporatesa two-stage ROM technique that enables rapid computation of UDmicrostructure nonlinear responses during UD CFRP structures simulation.The bottom-up multiscale modeling workflow for UD CFRP is explained inFIG. 44, which illustrates the length scale span in the modelingprocess. The UD CFRP microstructure at micron scale, characterization byUD RVE, provides microstructure information for UD lamina and UD couponat millimeter scale. Accurate UD structure responses are predicted usingphysical RVEs. From the microscale (the UD microstructure) to themacroscale (the UD Coupon FE mesh), the UD Coupon is modeled in a bottomup fashion. This framework can be extended to 3-scale composites,including UD, woven and woven laminate structures.

There has been considerable effort during the past decades inincorporating microstructure information to the macroscale model forperformance prediction. For example, one can model all microstructuredetails into a single model, but this computation is expensive due tothe fine mesh required. By deploying multiscale modeling techniques, themacroscale responses are predicted with physical microscale information.A hierarchical multiscale modeling approach as demonstrated by theMultiresolution Continuum Theory is one of them. In the MultiresolutionContinuum Theory, the microstructure information is implemented into themacroscale constitutive law to construct a hierarchical multiscalemodel, which captures microstructural effects, such as the effect ofinclusion size. Although this hierarchical multiscale modeling methodpreserves certain microstructure information, it does not provideexplicit microstructure evolution.

To capture microstructure responses, concurrent homogenization can bedeployed. One of the concurrent homogenization schemes is the FiniteElement square (FE²) approach. In the FE² approach, the macroscalegeometry is discretized with a Finite Element (FE) mesh. Materialresponses at all integration points are computed by solving RVEs thatare discretized with FE meshes. The FE² approach solves two sets of FEmeshes in a concurrent fashion, which results in high computationalcost.

To improve computational efficiency, various ROM approaches weredeveloped, such as Transformation Field Analysis, NonuniformTransformation Field Analysis and Proper Orthogonal Decomposition. Anewly proposed data-driven two-stage ROM approach, namelySelf-consistent Clustering Analysis (SCA), creates ROMs from highfidelity voxel mesh RVEs and simulates RVEs' elasto-plastic behaviorswith an effective RVE damage and failure. In the offline stage, athree-step approach is introduced: 1) data collection, such ascollecting strain concentration tensor for each voxel in the RVE; 2)Unsupervised learning, which classifies all voxel elements intodifferent clusters; 3) Generating cluster-wise interaction tensors. Inthe online prediction stage, cluster-wise strain responses areidentified by solving a discretized Lippmann-Schwinger equation. Aprevious publication shows that the SCA drastically reducescomputational expenses, and the accuracy is verified against a highfidelity Direct Numerical Simulation (DNS). Therefore, the three-steptwo-stage data-driven SCA method is a valuable tool in the ICME processfor modeling UD CFRP composites. An SCA flow chart for the solving theLippmann-Schwinger equation in one embodiment is shown in FIG. 66.

In this exemplary example, the ICME modeling framework for UD CFRPstructure performance prediction is presented. Under the framework, themacroscale UD model is discretized with an FE mesh. The UDmicrostructures are characterized by RVEs. RVEs are compressed into UDReduced Order Models (ROMs) and provide mechanical responses for allintegration points on-the-fly. The constitutive response of eachconstituent, e.g., fiber and matrix, is obtained from physical tests.The UD ROMs capture elastic and elasto-plastic responses of the UD CFRPthrough computing the RVE responses. Moreover, the UD non-linearity dueto matrix plasticity differentiates this work from previous efforts inCFRP ICME modeling. In those previous works, structural analysis is madeby assuming linear elastic material responses. The UD ROMs compute RVEresponses efficiently enough to replace the phenomenological anisotropicmaterial model and minimized material constants calibration effort.

The methodology developed can be popularized for predicting performanceof fiber reinforced polymer in general. Basic materials information, theexperimental procedures of the UD CFRP coupon off-axial tensile test andthe UD CFRP 3-pt bending test, ICME modeling process for the UD CFRP indetails, and results and experimental validation are discussed below.

Materials Information

In this section, the material properties obtained for A42 carbon fiberfrom DowAksa and thermoset epoxy resin from Dow chemical are provided.Material properties provided in this section are used through out themodeling work in this exemplary study.

The fibers elastic constants are given in Table 3-1 below. Fibers areassumed to behave elastically. The fiber direction compressive strengthis assumed to be one-fourth of the tensile strength, where the tensilestrength (TS) and compressive strength (CS) are assumed to be 4200 MPaand 1050 MPa, respectively. Ductile damage model for carbon fibers isassumed, as shown in Eq. (3-1), where d_(fiber)=0 means no damage andd_(fiber)=1 means fiber is damaged.

TABLE 3-1 Carbon Fiber Elastic Material Constants E₁₁ E₂₂ = E₃₃ G₁₂ =G₁₃ GPa 19.8 GPa 29.2 GPa G₂₃ v₁₂ = v₁₃ v₂₃ .92 GPa 0.28 0.32

$\begin{matrix}{d_{fiber} = \left( \begin{matrix}{0,{{\sigma_{11}} \leq {{TS}/{CS}}}} \\{1,{{\sigma_{11}} > {{TS}/{CS}}}}\end{matrix} \right.} & \left( {3\text{-}1} \right)\end{matrix}$

The epoxy resin elastic constants and tensile and compressive strengthsare given in Table 3-2. The tensile and compressive yielding curves aregiven in panels (a)-(b) of FIG. 45, respectively. Different tensile andcompressive behavior suggest that a paraboloid yielding criterion can beimplemented to capture such behavior.

TABLE 3-2 Material Constants of Epoxy Matrix E v σ_(T) σ_(C) 3803 MPa0.39 68 MPa 330 MPa

Experiments for the UD CFRP

Material characterizations provide a good understanding of the materialof interest. Various testing methods deliver the CFRP properties andprovide validation data for the prediction made by the ICME framework.In this work, two types of experiments are identified for examining thepredictivity of the proposed UD CFRP ICME framework: (1) UD coupon 10°off-axial tensile test and (2) UD 3-pt bending test.

UD CFRP Coupon 10° 10 Degree Off-Axial Tensile Test:

The UD CFRP coupon specimen is prepared through the following steps. Theedge of the UD plaque (with nominal fiber volume fraction of 50%) servedas reference for the determination of the angle for cutting the 10°off-axis orientation. The specimen head areas and the tab (wovenfiberglass in an epoxy resin) surfaces (with a length of 50 mm) wereprepared with grinding paper before applying a commercially availableacrylic adhesive. Metallic wires with a diameter of approximately 220 mwere used as spacers between specimen and the tab surfaces. The tabbingangle of about 16° was formed by grinding. The specimens were cut with awaterjet system to a nominal width (w) of 12.7 mm with a length of 210mm for a resulting aspect ratio of 9 in the gauge section of length1=120 mm, as shown in FIG. 46 (specimens 001-005). An abrasive grit sizeof 220 m and the lowest translation speed was chosen for minimalfabrication damage, based on preliminary studies for optimization of thewaterjet cutting parameters (i.e., nozzle diameter, pressure, speed,abrasive grit size, etc.).

The displacement-controlled tensile tests have been conducted on a servohydraulic testing machines at a quasi-static loading rate of 0.0167 mm/s(1 mm/min). The loading has been induced by the actuator located at thebottom of the test frames. Before each test, a precision steel block hadbeen used for rotational alignment of the actuator to reduceout-of-plane misorientation. The specimens were rigidly hydraulicallygripped with anti-rotation collars installed using diamond jaw surfacesand a pressure of 4 MPa. The gripping length on each side ranged between30 mm to 40 mm. The specimens have been aligned with specimen stops inthe grip.

All specimens were prepared for Digital Image Correlation (DIC)measurement with commercially available matte white spray paint,followed by applying matte black spray paint to create a random patternby the overspray method, as shown in panel (a) of FIG. 47, illustratinga field of view with speckle pattern on specimen with 1=120 mm, w=12.7mm. The region of interest (colored area) for strain measurement isshown in panel (b) of FIG. 47.

For stereo-DIC measurement, two 4.1 Mpx (2048 px×2048 px) cameras and 35mm fixed focal length lenses were used. The image acquisition rate was 2Hertz. The resolution was 60 m to 70 m per pixel and the size of thedark speckles was about 232 m (3.4 px), measured via the lineintersection method. The dark/bright ratio of the sample was nearly one(54:46). For data analysis, the chosen subset size was 15 px and thestep size was 6 px. The reference image has been taken at a force F=0 kNwhile the specimen has only been clamped by the top grip. For analysis,engineering strain has been calculated using a commercially availableDIC software package.

UD CFRP Hat-Section Dynamic 3-pt Bending Test:

The UD CFRP hat-section studied in the current work was molded with A42fibers, provided by DowAksa, and thermoset epoxy resin with fiber volumefraction of 50%. The geometry of the dynamic 3 point bending test samplewas shown in panel (a) of FIG. 48. The nominal thickness of thehatsection was 2.4 mm with about 0.2 mm thickness of each layer. Thehatsection sample was deformed in a hot compress and held for about 5minutes for curing. The hatsection was made with [0/60/−60/0/60/−60]slayup (noted as 0-60). In order to perform the test successfully, a backplate of the same layup and thickness was glued to the bottom of thehatsection sample with Betamate 4601 glue (Dow), as shown in panel (b)of FIG. 48.

The setup of the dynamic 3 point bending test was shown in panel (c) ofFIG. 48. The sample was slightly fixed to the lower roller (diameter 25mm) with tape in order to allow the rotation at the bottom. An impactorwith 25 kg mass and 100 mm outer diameter impacted the hatsection withinitial impact velocity of 4.66 m/s. The peak impactor acceleration andthe impactor force were recorded for comparison to the numericalpredictions.

UD CFRP ICME Modeling Process

In the ICME framework, the CFRP structure is modeled in a bottom-upfashion, as introduced in FIG. 44. At the microscale, UD CFRPmicrostructure is modeled as RVEs. RVEs are compressed into themicrostructure database, which contains ROMs for all RVEs. ROMs can befed into an arbitrary macroscale FE model composed of 3D stress stateelements, such as the brick element or the thick-shell element. The ROMsinteract with the macroscale model and enables a multiscale model withinformation exchange between the microscale and the macroscale. In thissection, the multiscale modeling workflow for the ICME process isprovided step by step.

ICME Multiscale Modeling Work Flow

For the illustration of ICME multiscale model setup, the UD CFRP couponspecimen geometry described above is used. The UD CFRP coupon FE modelmirrors the real UD CFRP coupon. It replicates the off-axial tensileexperiment performed as a one-to-one replica. The FE model contains all12 UD CFRP laminae. Each of these laminae is modeled as a singular layerof thick shell elements with 2 integration points in the thicknessdirection (Z direction), as shown in FIG. 49. For clarity, the FE modelhas been magnified by a factor of 2 in the thickness direction. The FEmodel contains 49,420 elements and 99,480 integration points. Themagnified region in FIG. 49 shows four selected thickshell elements andintegration points in each element. The UD CFRP microstructure, modeledby the UD CFRP RVE, is assigned to each integration point in order tocompute material responses under external loadings.

To illustrate the diversity of the microstructure database, two selectedintegration points, marked by the red box in FIG. 49, are furthermagnified for the underneath microstructure. FIG. 50 depicts how two UDRVEs from two neighboring integration points (as indicated by the dottedred box) are modeled using the microstructure database. The databasecontains four different RVE setups that could potentially be used forthe multiscale modeling process. On the right side of FIG. 50, theellipses around each RVE are used to indicate that there are hiddenneighboring RVEs on each side of the RVE due to the assumption ofperiodic boundary condition (PBC). The first setup assumes perfectbonding between the fiber and matrix, as well as for all laminae. Thesecond setup assumes a weak bond between laminae that can be modeled asa cohesive layer. The third setup assumes an interface region betweenfiber and matrix, which is modeled as non-zero thickness interphase. Thefourth setup assumes there is an interface region between fiber andmatrix and that there is weak bond between the laminae. In this work,only the first setup is incorporated due to assumption of perfectbonding.

The RVEs illustrated in FIG. 50 are compressed into UD CFRP ROMs throughthe three-step offline data compression stage. Details on the offlinestage will be provided in later subsections.

UD RVE Modeling

The cured UD CFRP lamina plaque is manufactured by Dow Chemical and thecross-section of the UD CFRP under microscope is shown in FIG. 51.Fibers are shown in lighter color and epoxy is shown in dark color. Thefibers are randomly dispersed in the epoxy resin matrix, with overallvolume metric fraction of 50%. The fibers are not a perfect circularshape, but rather a bean shape, as shown in FIG. 51. Assuming the beanshape fits into a ellipse domain, then the major axis of the ellipse ismeasured by the white line segment with length of 7 μm, as shown in themagnified view in FIG. 51. For convenience in modeling of fibers,circular shapes are assumed with a diameter of 7 μm.

RVEs are used to characterize the UD CFRP microstructure. In general,when the RVE is large enough, typically ten times of the fiber diameter,the random distribution of fiber does not affect the RVE responsessignificantly.

To generate the UD RVE, the fiber cross-section is simplified as acircular shape with a diameter of 7 μm, and the fiber is assumed to beperfectly straight. The cross-section of the UD RVE can be modeled in a2D fashion, where the Monte Carlo method is used to pack circlesrandomly in a 2D domain until the target fiber fraction is met. If partof any fiber lies outside four edges of the RVE, that part will reappearin the opposite directions to ensure periodic distribution of all fibersso the RVE complies with the PBC assumption. The generated 2D mesh isthen discretized by square pixels. The algorithm flow chart forgenerating a 2D mesh of the UD RVE is given in the FIG. 52 below. TheRVE algorithm discussed above is suitable for UD RVEs with fiber volumefraction around 50%. For higher volume fractions, specialized algorithmsare needed. This is beyond the scope of current work and is notdiscussed in detail. Finally, the 2D mesh is extruded by assigningthickness to all pixels to generate a 3D voxel mesh.

The generated UD RVE is given in FIG. 53. The RVE has a resolution of600 by 600 by 100 cubic voxels with voxel edge length of 0.14 μm. 93fibers are generated in the RVE. Fibers are assumed to be perfectlystraight and of circular shape.

To utilize UD RVE for compute stress responses on-the-fly in amacroscale FE model, the ROM technique is used to compress RVEs into amicrostructure database. The ROM process is given in following sections.

A DNS of RVE transverse tensile loading is also performed. The DNS isused to verify the efficacy of the ROM, which is supposed to produceaccurate results compared with DNS solution. Fiber and matrix propertiesare following data given above

Reduced Order Modeling of UD RVE

The aforementioned UD RVE contains significant DOFs for a single RVE rundue to the fine mesh resolution. To model the UD CFRP structure with UDRVEs, the computational cost is not affordable due to the costly RVEcomputation. Instead, the ROM technique is applied to the UD RVEs togenerate their ROMs. The ROM, in theory, will reduce the computationalcost significantly compared to RVE computation. The SCA approach is usedto generate the ROMs for the UD RVEs. All necessary derivations for SCAare provided in Appendix A. In this subsection, we will focus onillustrating the 3-step offline stage computation and the onlineprediction stage.

Offline: The offline stage starts with a high fidelity RVE discretizedby a voxel mesh. The strain concentration tensor A(x) links macroscopicstrain applied on the RVE to each voxel through the followingrelationship:

ε^(m)(x)=A(x):ε^(M)  (3-2)

where ε^(m) is the microscopic strain at any voxel in the RVE and ε^(M)is the applied macroscopic strain of the RVE. A(x) is the well-knownstrain concentration tensor. Under the Voigt notation, ε^(m) and ε^(M)are both 1 by 6 vectors. This means A(x) is a 6 by 6 matrix. A(x) can becomputed by applying six orthogonal loading conditions where ε^(M) hasonly one non-zero component at a time. This would allow A(x) to becomputed one column at a time, and six loading conditions can provideall 36 components of A(x).

Once A(x)s for each voxel are computed, unsupervised learning can beapplied to all A(x)s within the RVE to perform clustering. This processwill compress the original RVE made of many voxel elements into severalclusters. For UD RVE with fiber and matrix phases, fiber and matrixphases are decomposed separately. Number of clusters in fiber and matrixphases are denoted as K_(f) and K_(m), respectively. It is convenient todefine K_(f)+K_(m)=K. The clustering process for UD RVE setup one andtwo, as depicted in FIG. 50, is given in FIG. 54. The data compressionprocess is performed using an unsupervised learning method, such asK-means clustering.

Interaction tensors D_(IJ) must be computed between all cluster pairs.Once interaction tensors are computed, it is possible to solve forcluster-wise strain increments by solving the discretizedLippmann-Schwinger equation.

Online: The online stage involves solving the following residual form inEq. (3-3). Details of the SCA online stage is given in Appendix A.

r ^(I)=−Δε^(M)+Δε^(I)+Σ_(J=1) ^(K)[D ^(IJ):(Δσ^(J) −C ⁰:Δε^(J)],I=1,2,3,. . . ,K  (3-3)

where r^(I) is the residual of strain increment on each cluster. Theresidual can be minimized by first linearizing the Eq. (3-3) withrespect to Δε^(I) and solve for Δε^(I) that minimizes r^(I).

Note that in the present multiscale modeling scheme, ROMs are deployedon all integration points in the FE mesh of the composite laminatestructure. At each integration point, the macroscopic strain incrementΔε^(M) is provided by the FE solver and the cluster-wise stress andstrain responses are solved with Eq. (3-3). The homogenized RVE stressincrement, denoted as Δσ^(M), is returned back to the FE solver.

DNS Vs. Reduced Order Model

Two sets of ROMs of the UD RVE are generated with a different number ofclusters: 1) 2 clusters in the fiber phase (K_(f)=2) and 8 clusters inthe matrix phase (K_(m)=8); 2) 16 clusters in the fiber phase (K_(f)=16)and 16 clusters in the matrix phase (K_(m)=16).

To verify that the ROM will produce satisfactory results, a transversetensile test with maximum 0.02 strain magnitude is performed. The sameloading is applied to two ROMs as well. Stress and strain curves for allthree cases are plotted in FIG. 55. The DNS result is plotted with a 95%confidence interval. The results of first ROM and second ROM mentionedearlier are denoted as “SCA, 8 cluster” and “SCA, 16 cluster”respectively. From the results shown in FIG. 55, it is clear that bothROMs converge to the DNS solution. To save computational cost in theconcurrent multiscale modeling of the UD CFRP, the ROM with 8 clustersis used in the following structural level models for balanced accuracyand efficiency.

As mentioned above, a paraboloid yielding function is implemented toconsider different tensile and compressive behavior of the epoxy matrix.The matrix damage is modeled using a paraboloid epoxy damage model.

The UD ROM utilizes fiber and matrix to compute UD CFRP's materialresponses efficiently. In the UD structure modeling, an equivalentdamage model is applied to all ROMs to simulate the damage of RVEs. Thedamage of the RVE at each integration point will reduce load carryingcapacity of each element in the macroscale model. When the damageexceeds 0.5, the integration point will lose load carrying capacity. Themacroscale element will fail when all integration points in the elementhave lost load carrying capacity.

UD CFRP Coupon Off-Axial Tensile Simulation Model Setup

With all aforementioned information available, the UD CFRP Couponspecimen multiscale model is illustrated in FIG. 56. The appliedboundary condition is shown in FIG. 56. In the experiments, both tabsections are tightly clamped with a pressure of 4 MPa. The same clampingpressure is applied to both tab sections of the FE model. Moreover, thetwo surfaces of the upper tab section are fixed in the y direction, butthe two surfaces of the bottom tab section are allowed to move in the ydirection. A displacement towards the negative y direction is applied tothe bottom tab section so the coupon is extended, allowing the tensiletest to be repeated. One can see the FE coupon specimen model preservesmost of the experimental conditions. This aligns with the purpose ofICME modeling, where a real-world part is modeled and analyzed with asmany details as possible.

UD CFRP Dynamic 3-Point Bending Model Setup

The 3-point bending model is the second test case for validating theefficacy of the proposed framework of UD CFRP. A similar approach to theUD CFRP coupon model is adopted. The model of the UD CFRP hat-section isshown in FIG. 57, where a [0/60/−60/0/60/−60]_(s) layup is used. Asshown in FIG. 57, the 12 layers of the UD laminate structure are modeledexplicitly with thick shell elements. Such a setup allows for thecapturing failure of individual lamina under the effect of the impactor.

Up to this point, all necessary modeling steps of the UD CFRP structureare finished. The concurrent multiscale modeling framework is applied totest problems mentioned above. All test problems are modeled withrealistic geometries, hence the FE model utilizes the same boundaryconditions given in both tests. The experimental test data is used forvalidating the proposed UD CFRP concurrent multiscale modelingframework. The prediction capability of the framework is examinedagainst experiments conducted delicately. The results of both testproblems and associated discussion are given in the next section.

Results and Discussion

Concurrent simulation results of the off-axial coupon tensile test andthe 3-point bending test are discussed in this section. The same UD RVEshown above is used in both cases due to same fiber volume fraction.

UD CFRP Coupon Off-Axial Tensile Simulation

In this section, the concurrent multiscale 10° UD off-axial couponspecimen tensile simulation results are presented and compared againstexperimental results. A loading rate of 0.0167 mm/s gradually extendsthe coupon sample in the downwards direction. During the deformationprocess, a 10° stress band is formed as one can see in the FIG. 58. TheUD CFRP coupon off-axial tensile simulation is used to validate that theproposed multiscale modeling framework for UD CFRP material predicts thesystem behavior with good accuracy.

The comparisons between multiscale coupon simulation and the test dataare made for: 1) normal stress vs. normal strain; 2) y directiondisplacement; 3) y direction strain. Based on the comparisons, twopurposes are addressed: 1) To validate the multiscale multiscalemodeling framework for the UD CFRP material; 2) To demonstrate that thepresent UD CFRP multiscale model has considerable prediction capability.

For the FE model, the normal stress is computed using the reaction forcecomputed at the gauge cross-section area divided by the area of theoriginal coupon cross-section. The reaction force of the cross-sectionnear the top tab region of the coupon was recorded during thesimulation. The reaction force is then used to compute the normal stressof the multiscale model. The change of the gauge length was used tocompute normal strain. Normal stress versus strain of the multiscalemodel is plotted as blue dots in FIG. 58.

Comparing the stress and strain curves from the multiscale model and theexperimental data, a good match is observed. The prediction has sametrend as the experimental data, as shown in Table 3-3. The predictedmaximum stress is 404.81 MPa, which is close to 395.64 MPa reported fromthe experimental data. In addition, the maximum strain predicted bymodel is 0.011, which again is in a good match with experimentalmeasured value of 0.012. Both predicted quantities of interest arewithin 10% deviation from experimental measurements, meaning the UD CFRPconcurrent multiscale model is validated against the experimental data.

TABLE 3-3 Predicted maximum normal stress and strain of the off-axial UDcoupon sample Maximum Normal Stress Maximum Normal (MPa) StrainExperiment 395.64 0.012 Prediction 404.81 0.011 Difference 2.32% 8.33%

FIG. 59: Contour of a) Y displacement and b) Y strain field. The applieddisplacement on prediction and DIC is 0.9031 mm. In the displacementplots a), two black arrows measure the vertical distance between fringesfrom −0.250 mm and −0.700 mm and the difference is 3.95%. In the grayscale strain contour b), the predicted strain field is comparable to theDIC one. The difference the predictions and the DIC images is caused bymicrostructure variations in the real UD CFRP material, which can causestrain concentration in the real sample.

The y direction displacement and strain fields are validated withexperimental results, as shown in panels (a)-)(b) of FIG. 59,respectively. The black arrows in panel (a) of FIG. 59 represent thevertical distance between fringes from −0.250 mm and −0.700 mm. Theprediction made by the multiscale model is 80.26 mm, with a 3.95%difference to the DIC measurement of 83.57 mm. As shown in panel (b) ofFIG. 59, the y direction strain yield contour predicted is quite similarwith the DIC measurement. Both strain field contours show a clear bandacross the middle of the coupon. The DIC strain field is highlynon-uniform due to unavoidable microstructure variation duringmanufacturing process. Still, the results are encouraging since thepredicted displacement and strain fields agree with experimentalresults.

Moreover, the predicted crack formation and the actual coupon crackformation are depicted in panels (a)-)(b) of FIG. 60, respectively. Inboth panels (a)-(b) of FIG. 60, the crack propagates all the way acrossthe coupon gauge section. The pattern of predicted crack is not an exactreplica of the experimental results because the numerical model has yetto take local microstructure variation into account, as depicted in FIG.51. In the future work, the local microstructure variation will beconsidered in the modeling process to address the uncertainty effect onthe macroscopic performance. Nonetheless, the ICME model is able toprovide an accurate prediction of the maximum stress and strain of thecoupon sample, as reported in Table 3-3, as well as a prediction of thefailure pattern. The prediction made by the ICME model is consistentwith the experimental observation, which provides confidence for thepredictivity of the proposed multiscale modeling scheme for the ICMEprocess.

The ICME multiscale modeling scheme has been validated by theexperimental data discuss above. However, the capability of the proposedmultiscale modeling scheme is beyond providing accurate prediction ofthe macroscale quantities. It also provides detailed microstructureevolution of UD CFRP structure for studying the root cause of thefailure in the UD coupon.

FIG. 61 illustrate the local UD CFRP damage process using threesnapshots. The magnified views of the region marked in red are shown inpanels (a)-(c) of FIG. 61 at y displacement of 1.40 mm, 1.41 mm, and1.42 mm. The UD RVEs representing elements marked in black are shown inpanels (d)-(f) of FIG. 61. In panel (e) of FIG. 61, all three RVEs areexperiencing matrix damage, suggesting the potential cause of couponfailure. In panel (b) of FIG. 61, the bottom element has a stresscontour turning to blue-green from green, suggesting reduced loadcarrying capacity. This is due to aggravated matrix failure of themiddle and bottom RVEs as shown in panel (e) of FIG. 61. In panel (c) ofFIG. 61, the middle and bottom elements have been marked as failed dueto the loss of load carrying capacity, whereas the top one is stillcapable of carrying load. However, as shown in panel (f) of FIG. 61, thetop RVE is also experiencing severe matrix damage, which means this RVEwill fail shortly. The UD RVE microstructure evolution in FIG. 61provides valuable information in the understanding of microscale damageprocess. Moreover, the UD RVE stress and damage evolution are capturedsimultaneously as the macroscopic UD coupon simulation. In the future,in-situ monitoring techniques can be combined with the existing modelingcapability to further validate microstructure failure process of the UD

Using detailed microstructure evolution information illustrated in FIG.61, a path towards UD CFRP design is discovered. For example, one candesign matrix strength to avoid matrix damage at small loadingmagnitude. Or, one can design interphase properties and incorporate theinterphase region into the model to examine the interphase effect andthe delamination behavior of the UD CFRP.

UD 3-Point Bending Model

To further illustrate the efficacy of the concurrent scheme, the UD3-point bending model concurrent simulation is also performed. Thefractured hat-section of the simulation and experiment is shown in FIG.64. The UD 3-point bending simulation is used to examine theapplicability of the proposed scheme to an arbitrary UD CFRP structure.Two quantities are chosen to make the comparison: peak load on theimpactor and the peak impactor acceleration. Due to severe vibration ofthe hat-section in the 3-point bending simulation, the matrix has beensimplified as an elastic material with brittle failure, where thefailure strength is set to compression strength as defined in Table 3-2.

After the impact, the hat-section from the numerical prediction andexperimental result are plotted in panels (a)-(b) of FIG. 62,respectively. The yellow ellipse marked damaged zones on thehat-section. It can be observed that the impact push material inwards,cause delamination of the hat-section on both side-walls. The similartrend observed in the numerical model and the experimental resultfurther suggests the multiscale model can provide a good prediction ofthe failure pattern on the UD structure. The ICME process can be appliedto predict the responses of a UD structure under complex loadingconditions.

In addition to the good match of the UD structure deformation andfailure, quantitative comparisons for peak load on the impactor and peakacceleration of the impactor are reported in Table 3-4 with comparisonwith the experimental data. A reasonable match between prediction andexperimental data can be seen. Specifically, the relative differences ofpeak load and peak impactor acceleration are 8.21% and 2.82%,respectively. Both predictions are within 10% deviation from theexperimental data, providing confidence that the macroscopic performanceindices can be predicted with the ICME framework.

TABLE 3-4 Impactor peak load and acceleration Peak Impactor Peak Load(N)Acceleration (m/s²) Experiment 10328 0.39 Prediction 9480 0.379Difference 8.21% 2.82%

In panel (a) of FIG. 62, the failure of the hat-section is depicted atthe macroscale. Underneath the complex macroscopic structure evolution,local RVE responses are captured in a concurrent fashion and provideextra information for the microscale evolution in the hat-section. Toillustrate the von Mises stress evolution in the UD microstructure, RVEsrepresenting three columns of elements shown in FIG. 63 are visualized.The three columns of elements are shown in the magnified view on theright of FIG. 62. Each column contains 12 layers of UD laminae, whereeach lamina is visualized with 1 UD RVE. The fiber orientation on eachUD laminate is color coded as shown in FIG. 63. All UD RVEs are alignedwith the fiber orientation as suggested by the color code, following thelamina orientation. All UD layers are counting from layer 1 to layer 12in the top-down fashion.

The hat-section deformation and von Mises stress contour under theimpactor are shown in the upper half of FIG. 64. Three differentsnapshots were taken with the impactor and supports hidden at: uponimpact; Impactor displacement of 4.85 mm; Impactor displacement of 6.85mm. In the upper half of panel (a) of FIG. 64, the impactor contacts thehat-section and caused immediate stress concentration at the contactregion. In panels (b)-(c) of FIG. 64, it can be observed that the topand two sidewalls of the hat-section bend inwards after the hat-sectioncontacts the impactor. The impactor pushes the middle of the UD CFRPhat-section inwards, simulating the scenario where the compositestructure is under the external loading condition.

Corresponding RVE von Mises stress contours are shown in the lower halfof panels (a)-(c) of FIG. 64. It can be seen that UD RVEs can be used toinvestigate the inter-laminate stress distribution of the UDmicrostructure. Due to different fiber orientations in the hat-sectionlaminate structure, the von Mises stress magnitude varies from layer tolayer across the thickness direction, which might lead to earlierfailure for those layers with larger stress magnitude. The RVE plotsshown in panel (b) of FIG. 64 suggested that the second and third inlocation 1 has much higher stress magnitude comparing with those inother layers. In panel (c) of FIG. 64, those two layers have reachedfailure as the stress magnitude are zero. This suggested that themultiscale modeling has great potential in predicting microstructuralevolution in a complex UD CFRP laminate structure. Such information canbe used to examine various UD CFRP laminate designs using the virtualverification capability to examine macro and micro material performance.This can assist the CFRP design process by eliminating designs thatunder-perform certain performance indices, or designs with undesiredmicrostructural evolution pattern.

For UD CFRP materials, understanding the microstructure failuremechanism provides valuable information in improving CFRP design.Similar to the UD CFRP coupon model, the microstructure evolution of thehat-section is captured and illustrated in the three-snapshot view inFIG. 65 with different impactor displacement. In FIG. 65, two UD CFRPRVEs representing two marked regions are visualized. It can be seen thatthe RVE in the upper layer begins to damage while the RVE in the lowerlayer stays intact when d_(y)=5.91 mm, as shown in panel (a) of FIG. 65.When the upper layer fails, the neighboring region will collapse towardsthe newly formed empty region and the neighbor elements will compressthe lower layer. Soon, the lower layer fails as well when d_(y)=6.13 mm,as shown in panel (b) of FIG. 65. Eventually, both layers fails as shownin panel (c) of FIG. 65. This three snapshots illustrate the ability ofinvesting detailed microstructure evolution of UD CFRP for a structurallevel simulation. Such information can provide guidance for design ofCFRP structure against local damage to improve the structuralperformance.

In this subsection, a one-to-one replica of the UD CFRP Coupon 10°off-axial tensile test multiscale simulation and the UD CFRP hat-section3-pt bending test are resolved using the proposed multiscale modelingframework. Both models are validated against experimental data forvalidation of the framework. Predictions made by the numericalcounterparts are all within 10% difference compared with experimentaldata. The agreement shows the current work can be used for prediction ofother UD CFRP laminate structure. Moreover, the concurrent capture ofmicrostructure evolution provides microstructure evolution history forany location in the FE model of the UD laminate. This allows researchersto look at the detailed microstructure evolution, which is hard tocapture experimentally, for the cause of failure at the structurallevel. New microstructure can be then designed to sustain the loadingand improve the overall structural performance.

Briefly, the exemplary study introduces a predictive and efficient ICMEmultiscale modeling framework for UD CFRP materials. The main workflowof the framework is explained in detail, and experimental validationsare provided. The predictive framework links UD CFRP microstructures tostructural level models for accurate prediction of the structuralperformance. Two sample cases studies, the UD off-axial tensile test andthe UD hat-section dynamic three-point bending test, are presented usingthe proposed ICME modeling framework. The predicted performance indicesare validated against experimental data confirming a good agreement.Microstructure evolution in the UD structure is captured by UD CFRPRVEs, which reveal microstructural evolution, including stress contourand matrix damage. The ICME framework is general and can be applied toother Fiber Reinforced Polymer (FRP) systems, such as glass fiberreinforced polymer, for structure performance prediction. Along with themicrostructure information, the work presented in this example shouldprovide guidance to existing experimental based composites designworkflows to accelerate the design process.

Future work of the present multiscale modeling framework for UD CFRPincludes: 1) Incorporation of the interphase in the UD RVE forfiber-matrix debonding; 2) Consideration of microstructureuncertainties, such as fiber misalignment and fiber volume fraction, forquantitative measurement of the microstructure effect. 3) Extension toother composite material systems.

Example 4 Data-Driven Self-Consistent Clustering Analysis ofHeterogeneous Materials with Crystal Plasticity

To analyze complex, heterogeneous materials, a fast and accurate methodis needed. This means going beyond the classical finite element method,in a search for the ability to compute, with modest computationalresources, solutions previously infeasible even with large clustercomputers. In particular, this advance is motivated by compositesdesign. Here, we apply similar principle to another complex,heterogeneous system: additively manufactured metals.

The complexities and potential benefits of metal additive manufacturing(AM) provide a rich basis for development of mechanistic materialmodels. These models are typically based on finite element modeling ofmetal plasticity. Powder-bed AM uses a high-power laser or electron beamto melt powder layer-by-layer to produce freeform geometries specifiedby 3D model files. This approach removes the need for special tooling,allowing for rapid and customized part and product realization. Itintroduces new possibilities for topological and material optimization,but these tasks require a high degree of knowledge and ability to applythat knowledge, viz. control the build conditions sufficiently. Theprocess involves intense and repeated localized energy input, whichresults in inhomogeneous, anisotropic, location-dependent materialproperties with complex microstructures.

The perpetual challenge in multiscale modeling is predicting macroscopicbehavior from microstructure conformation and properties in both anefficient and accurate manner. Analytical approaches such as the rule ofmixtures and other micromechanics methods are very efficient but loseaccuracy particularly when dealing with irregular morphologies andnonlinear properties. In contrast, direct numerical simulations (DNS),offer high accuracy at the expense of prohibitive computational costs tothe point where they are inapplicable to concurrent simulations formaterial design. Recently, data mining has been introduced into themechanics community to address the limitations of DNS and analyticalmethods.

In general, data mining is a computational process of discoveringpatterns in large data sets. Once extracted, these patterns can be usedto predict future events. Machine learning methods are the technicalbasis for data mining, such as clustering and regression methods.Recently, data mining has also been applied to the modeling ofheterogeneous materials. As a start, a raw dataset for learning isusually generated from a priori numerical simulations or informed byexperiments. Depending on the type of the raw dataset, currentdata-driven modeling methods can be mainly divided into macroscopic andmicroscopic approaches.

In macroscopic approaches, the input data are usually materialproperties of each constituent, loading conditions and statisticaldescriptors that represent the geometry of the microstructures, whilethe output data are macroscopic mechanical properties from directnumerical simulations (DNS). However, the accuracy and smoothness of theprediction of macroscopic approaches is limited by a lack of microscopicinformation. For example, the localized plastic strain fields, criticalfor plasticity and damage prediction theories, cannot be wellrepresented by their field averages.

To address this problem, microscopic approaches collect data at eachdiscretization point in the DNS. Two methods for making predictionsbased on gathered local data are worth highlighting: (1) non-uniformtransformation field analysis (NTFA) and (2) variants of the properorthogonal decomposition (POD) method. For both approaches, thepredictions under a loading condition are obtained by linearcombinations of a finite number of RVE modes from previously completedsimulations under various load conditions. Linear combination ofeigenmodes is well established, but extra effort is required for theinterpolation for nonlinear materials. For NTFA, specific evolution lawsof internal variables have to be assumed for each mode of the inelasticfield. For POD-based methods, extensive simulations a priori are neededin order to guarantee the robustness of the interpolation underarbitrary loading conditions. However, this still results in an overalldecrease in computational cost.

One of the objectives of this exemplary study is to present agrain-level crystal plasticity model to capture local microstructuressuch as those that occur in AM, e.g., voids and columnar grains. In oneembodiment, a recently introduced material modeling approach, based onthe theories of data mining and originally developed for composites, isused to vastly increase the speed of these simulations. This allows forhigher detail or larger regions of interest, both of which are desirablefor predicting damage and fatigue initiation within a part.

The exemplary study is based on a two-stage approach that usesclustering and subsequent analysis of deformation and can account forheterogeneous material behavior with high accuracy and speed, which iscalled SCA. It is a data-driven method designed to reduce thecomputational degrees of freedom (DOFs) required for predictingmacroscopic behaviors of heterogeneous materials, while localinformation is retained based in part on clustering near features thatinduce large stress gradients. The basic idea is to solve theequilibrium equation, not at every material point, but on clusters ofmaterial points with similar mechanical responses by assuming that localvariables of interest (e.g., elastic strain, plastic strain and stress)are uniform in each of these clusters. The two stages of the method are:offline (training or cluster) and online (prediction).

During the offline stage, material points were grouped into clustersusing data mining techniques (such as k-means clustering) based onmechanical similarity. To conduct the online computation, theequilibrium equation was written in an integral form using the Green'sfunction, known as the Lippmann-Schwinger equation. This equation wassolved for each cluster using a self-consistent scheme to ensure theaccuracy, where the reference stiffness was updated iteratively to beconsistent with the macroscopic effective stiffness. The major advantageof SCA is that the DOFs are greatly reduced compared to DNS whileretaining both local and global response information.

Here we extend this method to be applicable to crystal plasticity (CP),a class of computational plasticity problems specifically formulated tocapture the deformation mechanics of crystalline solids, based on thematerial microstructure. Anisotropic material models such as CP havebeen derived for both macroscale problems, such as predicting earingduring deep drawing, and microscale problems, such as the deformation ofnanowires.

SCA Framework

Offline Stage: Mechanistic Material Characterization

Grouping material points with similar mechanical behavior into a singlecluster is performed by domain decomposition of material points usingclustering methods. First, the similarity between two material points ismeasured by the strain concentration tensor A(x), which is defined as

ε^(micro)(x)=A(x):ε^(macro)inΩ,  (4-1)

where ε^(macro) is the elastic macroscopic strain corresponding to theboundary conditions of the Representative Volume Element (RVE), andε^(micro)(x) is the elastic local strain at point x in the microscaleRVE domain Ω. For a 2-dimensional (2D) model, A(x) has nine independentcomponents, requiring a set of elastic direct numerical simulations(DNS) under three orthogonal loading conditions to uniquely define. Fora 3-dimensional (3D) model, A(x) has 36 independent components which aredetermined by DNS under six orthogonal loading conditions. Once thestrain concentration tensor is computed, it is independent of theloading conditions for a linear elastic material, and its Frobenius normis an invariant under coordinate transformation.

For overall responses of nonlinear plastic materials, we havedemonstrated that the elastic strain concentration tensor is a goodoffline database. However, if the local response is of more interest,the elastic strain concentration tensor often does not provide highenough cluster density near the high strain concentration region. Inpolycrystalline material with crystal plasticity, all the crystalsdeform uniformly in the elastic regime, providing no effective data forcomputing the strain concentration tensor. In these cases, one canchoose other types of material responses to construct the offline data,and thus achieve adequate resolution at the region of interest. Forexample, we choose the plastic strain tensor from DNS calculations forclustering when local plasticity information is required, such as forpredicting fatigue initiation. We show how the choice of differentmaterial responses affect overall response prediction.

The k-means clustering method is used to group data points based on agrouping metric. For present, let us consider this to be the strainconcentration tensor A(x). Since all the material points in a clusterare assumed to have the same mechanical response, the number of thedegrees of freedom is reduced from, e.g., the number of elements in aFEM simulation to the number of clusters. Note that clusters are formedbased on the strain concentration tensor and thus do not need to bespatially adjacent to each other.

A primary assumption associated with the domain decomposition is thatany local variable β(x) is uniform within each cluster. Globally, thisis equivalent to having a piece-wise uniform profile of the variable inthe RVE:

β(x)=Σ_(I=1) ^(k)β^(I)χ^(I)(x),  (4-2)

where β^(I) is the homogeneous variable in the Ith cluster, and k is thetotal number of clusters in the RVE. The domain of the Ith cluster Ω^(I)is distinguished by its characteristic function χ^(I)(x), which isdefined as

$\begin{matrix}{{\chi^{I}(x)} = \left\{ \begin{matrix}1 & {{{if}\mspace{14mu} x} \in \Omega^{I}} \\0 & {otherwise}\end{matrix} \right.} & \left( {4\text{-}3} \right)\end{matrix}$

This piecewise uniform approximation in Eq. (4-2) enables us to reducethe number of degrees of freedom for the Lippmann-Schwinger equation,which is solved in the following online stage. After the domaindecomposition based on a prior DNS, the remaining task in the offlinestage is to pre-compute the interaction tensors between all theclusters.

In the discretized/reduced Lippmann-Schwinger equation, we can utilizethe piecewise uniform assumption to extract the interaction tensorD^(IJ), which represents the influence of the stress in the Jth clusteron the strain in the Ith cluster. In an RVE domain Ω with periodicboundary conditions, the interaction tensor can be written as aconvolution of the Green's function and the characteristic functionsdefined in Eq. (4-3):

$\begin{matrix}{{D^{IJ} = {\frac{1}{c^{I}{\Omega }}{\int_{\Omega}{\int_{\Omega}{{\chi^{I}(x)}{\chi^{J}\left( x^{\prime} \right)}{\Phi^{0}\left( {x,x^{\prime}} \right)}{dx}^{\prime}{dx}}}}}},} & \left( {4\text{-}4} \right)\end{matrix}$

where c^(I) is the volume fraction of the Ith cluster and |Ω| is thevolume of the RVE domain. Φ⁰(x,x′) is the fourth-order periodic Green'sfunction associated with an isotropic linear elastic reference materialand its stiffness tensor is C⁰. Specifically, this reference material isintroduced in the online stage as a homogeneous media to formulate theLippmann-Schwinger integral equation. With the periodicity of the RVE,Φ⁰(x,x′) takes the following form in the Fourier space,

$\begin{matrix}{{{{\hat{\Phi}}^{0}(\xi)} = {{\frac{1}{4\mu^{0}}{{\hat{\Phi}}^{1}(\xi)}} + {\frac{\lambda^{0} + \mu^{0}}{\mu^{0}\left( {\lambda^{0} + {2\mu^{0}}} \right)}{{\hat{\Phi}}^{2}(\xi)}}}},{with}} & \left( {4\text{-}5} \right) \\{{{\hat{\Phi}}_{ijkI}^{1}(\xi)} = {\frac{1}{{\xi }^{2}}\left( {{\delta_{ik}\xi_{j}\xi_{l}} + {\delta_{il}\xi_{j}\xi_{k}} + {\delta_{jt}\xi_{i}\xi_{k}} + {\delta_{jk}\xi_{i}\xi_{l}}} \right)}} & \left( {4\text{-}6} \right) \\{{{{\hat{\Phi}}_{ijkl}^{2}(\xi)} = {- \frac{\xi_{i}\xi_{j}\xi_{k}\xi_{l}}{{\xi }^{4}}}},} & \left( {4\text{-}7} \right)\end{matrix}$

where ξ is the coordinate in Fourier space corresponding to x in realspace, and δ_(ij) is the Kronecker delta function. λ⁰ and μ⁰ are Laméconstants of the reference material. The expression of {circumflex over(Φ)}_(ijkl) ⁰(ξ) is not well defined at frequency point ξ=0. However, byimposing the boundary conditions for deriving the Green's function, auniformly distributed polarization stress field will not induce anystrain field inside the RVE. As a result, we have

{circumflex over (Φ)}_(ijkl) ⁰(ξ=0)=0.  (4-8)

Based on Eq. (4-5), the convolution term in the spatial domain in Eq.(4-4) can be translated into a direct multiplication at each point inthe frequency domain using a Fourier transformation,

Φ _(J) ⁰(x)=ω_(Ω)χ^(J)(x′)Φ⁰(x,x′)dx′=

⁻¹({circumflex over (χ)}(ξ){circumflex over (Φ)}⁰(ξ)).  (4-9)

As we can see from Eq. (4-6), {circumflex over (Φ)}¹(ξ) and {circumflexover (Φ)}²(ξ) are independent of the material properties, so that theycan be computed once, in the offline stage. If the reference material ischanged in the self-consistent scheme during the online stage only thecoefficients relating to the reference Lamé constants in Eq. (4-5) needto be updated. For RVEs with microstucture size close to the RVE size oreven with a connected microstructure network, such as a woven composite,a correction of D^(IJ) is needed to satisfy the boundary conditions onthe RVE.

Currently, we have applied the SCA offline stage to 2D and 3D materialswith uniform (regular hexahedral or “voxel”) meshes, so that the FastFourier transformation (FFT) method can be used for efficientlycomputing Eq. (4-9). Although the domain decomposition is based on aspecific selection of properties for each material phase in the offlinestage, the same database can be used for predicting responses for newcombinations of material constituents in the online stage.

Online Stage: Self-Consistent Lippmann-Schwinger Equation

The equilibrium condition in the RVE can be rewritten as a continuousLippmann-Schwinger integral equation by introducing a homogeneousreference material,

Δε(x)+∫_(Ω)Φ⁰(x,x′):[Δσ(x′)−C ⁰:Δε(x′)]dx′−Δε ⁰=0,  (4-10)

where Δε⁰ is the far-field strain increment controlling the evolution ofthe local strain. It is uniform in the RVE. The reference material isisotropic and linear elastic. Its stiffness tensor C⁰ can be determinedby the two independent Lamé parameters λ⁰ and μ⁰,

C ⁰ =f(λ⁰,μ⁰)=λ⁰ I⊗I+μ ⁰ II.  (5-11)

where I is the second-rank identity tensor, and II is the symmetric partof the fourth-rank identity tensor. The strain and stress increments areΔε(x) and Δσ(x). By averaging the incremental integral equation, Eq.(4-10), in the RVE domain Ω, we have

$\begin{matrix}{{{\frac{1}{\Omega }{\int_{\Omega}{\Delta\;{ɛ(x)}{dx}}}} + {{\frac{1}{\Omega }\left\lbrack {\int_{\Omega}{{\Phi^{0}\left( {x,x^{\prime}} \right)}{dx}}} \right\rbrack}{\text{:}\;\left\lbrack {{\Delta\;{\sigma\left( x^{\prime} \right)}} - {C^{0}\text{:}\mspace{14mu}{{\Delta ɛ}\left( x^{\prime} \right)}}} \right\rbrack}{dx}^{\prime}} - {\Delta\; ɛ^{0}}} = 0.} & \left( {4\text{-}12} \right)\end{matrix}$

Imposing by the boundary conditions for deriving the Green's function,Eq. (4-8) can be equivalently written as

∫_(Ω)Φ⁰(x,x′)dx=0.  (4-13)

Substituting Eqs. (4-13) into (4-12) gives

$\begin{matrix}{{{\Delta ɛ^{0}} = {\frac{1}{\Omega }{\int_{\Omega}{\Delta{ɛ(x)}dx}}}},} & \left( {4\text{-}14} \right)\end{matrix}$

which indicates that the far-field strain increment is always equal tothe ensemble averaged strain increment in the RVE. In order to solveΔε(x) in the integral equation, Eq. (4-10), constraints are needed fromthe macroscopic boundary conditions. The macro-strain constraint can bewritten as

$\begin{matrix}{{{\frac{1}{\Omega }{\int_{\Omega}{\Delta{ɛ(x)}dx}}} = {{\Delta\;{\overset{\_}{ɛ}}^{macro}{or}\;{\Delta ɛ}^{0}} = {\Delta\;{\overset{\_}{ɛ}}^{macro}}}},} & \left( {4\text{-}15} \right)\end{matrix}$

where Δε ^(macro) is the macroscopic strain increment applied on theRVE. Similarly, the macro-stress constraint can be related to themacroscopic stress σ ^(macro),

$\begin{matrix}{{\frac{1}{\Omega }{\int_{\Omega}{{\sigma(x)}{dx}}}} = {\overset{\_}{\sigma}}^{macro}} & \left( {4\text{-}16} \right)\end{matrix}$

For more general cases, mixed constraints (e.g., strain constraint inthe 11-direction and stress constraints for the rest) can also beformulated.

As the full-field calculations (e.g., FFT-based method) of thecontinuous Lippmann-Schwinger equation may require excessivecomputational resources, we will perform the discretization of theintegral equation based on the domain decomposition in the offlinestage. With the piecewise uniform assumption in Eq. (4-2), the number ofdegrees of freedom and the number of internal variables in the newsystem can be reduced. After decomposition, the discretized integralequation of the Ith cluster is:

Δε^(I)+Σ_(J=1) ^(k) D ^(IJ):[Δσ^(J) −C ⁰:Δε^(J)]−Δε⁰=0,  (4-17)

where Δε^(J) and Δσ^(J) are the strain and stress increment in the Jthcluster. The interaction tensor D^(IJ) is defined in Eq. (4-4), which isrelated to the Green's function of the reference material. Afterdiscretization, the far field strain is still equal to the averagestrain in the RVE,

Δε⁰=Σ_(I=1) ^(k) c ^(I)Δε^(I).  (4-18)

The macroscopic boundary conditions also require discretization. Forinstance, the discrete form of the macro-strain constraint can bewritten as

Σ_(I=1) ^(k) c ^(I)Δε^(I)=Δε ^(macro) or Δε⁰=Δε ^(macro)  (4-19)

In the new reduced system, the unknown variables are the strainincrements in each cluster Δε^(I). Significantly fewer clusters than FEnodes means that the ROM is much faster to solve. In a general case suchas plasticity, the stress increment Δσ^(J) is a nonlinear function ofits strain increment Δε^(J), and Newton's method is used to solve thenonlinear system iteratively for each increment.

The solution of the continuous Lippmann-Schwinger equation (4-10) isindependent of the choice of the reference material C⁰. This is becausethe physical problem is fully described by the equilibrium condition andthe prescribed macroscopic boundary conditions. However, the equilibriumcondition is not strictly satisfied at every point in the RVE for thediscretized equations because of the piecewise uniform assumption, andthe solution of the reduced system depends on the choices of C⁰. Thisdiscrepancy can be reduced by increasing the number of clusters used, atthe cost of increased computational cost due to the increased degrees offreedom.

To achieve both efficiency and accuracy, we propose a self-consistentscheme in the online stage, which retains accuracy with fewer clusters.In the self-consistent scheme, the stiffness tensor of the referencematerial, C⁰ is approximately the same as the homogenized stiffnesstensor C,

C ⁰ →C.  (4-20)

Material non-linearity generally makes it impossible to determine anisotropic C⁰ exactly matching C. Here we propose two types ofself-consistent schemes to approximate Eq. (4-20): 1) linear regressionof average strain increment Δε and stress increment Δσ (orregression-based scheme) and 2) isotropic projection of the effectivestiffness tensor C (or projection-based scheme).

Regression-Based Self-Consistent Scheme

In the regression-based scheme, the self-consistent scheme is formulatedas an optimization problem: the goal is to find an isotropic C⁰ thatminimizes the error between the predicted average stress increments. Theinputs of the regression algorithm are the average strain increment aand stress increment Δσ, which are computed as

Δε=Σ_(I=1) ^(k) c ^(I)Δε^(I) and Δσ=Σ_(I=1) ^(k) c ^(I)Δσ^(I)  (4-21)

The objective of the regression-based scheme is to find the λ⁰ and μ⁰ ofthe reference material by computing

$\begin{matrix}{\left\{ {\lambda^{0},\mu^{0}} \right\} = {\underset{\{{{\lambda\;\prime},{\mu\;\prime}}\}}{\arg\min}{{{{\Delta\overset{¯}{\sigma}} - {{f\left( {\lambda^{\prime},\mu^{\prime}} \right)}\text{:}\mspace{14mu}\Delta\overset{¯}{ɛ}}}}^{2}.}}} & \left( {4\text{-}22} \right)\end{matrix}$

where ∥Z∥²=Z:Z for an arbitrary second-order tensor Z. The functionƒ(λ′,μ′) can be expressed as

ƒ(λ′,μ′)=λ′I⊗I+μ′II.  (4-23)

where I is the second-rank identity tensor, and II is the symmetric partof the fourth-rank identity tensor. Equivalently, the cost functiong(λ′,μ′) of the optimization problem can be written as

g(λ′,μ′)=∥Δσ−ƒ(λ′,μ′):Δε∥².  (4-24)

The optimum point is found via the respective partial derivatives of thecost function,

$\begin{matrix}{{{\frac{\partial g}{\partial{\lambda\prime}}{_{\lambda^{0},\mu^{0}}{= {0\;{and}\;\frac{\partial g}{{\partial\mu}\;\prime}}}}_{\lambda^{0},\mu^{0}}} = 0},} & \left( {4\text{-}25} \right)\end{matrix}$

which forms a system of two linear equations in terms of the Laméconstants. The system always has a unique solution except under apure-shear loading condition, where λ⁰ is under-determined. In thiscase, the value of λ⁰ is not updated. Additionally, g(λ⁰,μ⁰) vanisheswhen the effective macroscopic homogeneous material is also isotropiclinear elastic.

Although this scheme does not require computing C explicitly, it has twomain drawbacks. First, the optimization problem is under-determined forhydrostatic and pure shear loading conditions, forcing one of the twoindependent elastic constants to be assumed. Second and more important,the modulus of the optimum reference material may be negative forcomplex loading histories within a concurrent simulation, which isdeleterious to the convergence of the fixed-point method.

Projection-Based Self-Consistent Scheme

To avoid the difficulties of the regression-based scheme, we presentanother self-consistent scheme based on isotropic projection of theeffective stiffness tensor C. Through the homogenization, the effectivestiffness tensor C of the RVE can be expressed as

C=Σ _(I=1) ^(k) c ^(I) c ^(I) C _(alg) ^(I) :A ^(I),  (4-26)

where C_(alg) ^(I) is the algorithm stiffness tensor of the material inthe Ith cluster and is an output of the local constitutive law for thecurrent strain increment in the cluster,

$\begin{matrix}{C_{alg}^{I} = \frac{\partial{\Delta\sigma}^{I}}{\partial{\Delta ɛ}^{I}}} & \left( {4\text{-}27} \right)\end{matrix}$

The strain concentration tensor of the Ith cluster A^(I) relates thelocal strain increment in the Ith cluster Δε^(I) to the far-field strainincrement Δε⁰,

Δε^(I) =A ^(I):Δε⁰,  (4-28)

which can be determined by first linearizing the discretized integralequation (4-17) using C_(alg) ^(I) and then inverting the Jacobianmatrix. Since C is only required for the self-consistent scheme, thecalculation of C can be performed once, after the Newton iterations haveconverged, to save computational cost.

For a 3D problem, the stiffness tensor of the isotropic referencematerial C⁰ can be decomposed as

C ⁰=(3λ⁰⁺2μ⁰)J+2μ⁰ K,  (4-29)

where the forth-rank tensors J and K are defined as

J=⅓(I⊗I) and K=II−J.  (4-30)

Since the two tensors are still orthogonal, we have

J::K=0,J::J=1,K::K=5.  (4-31)

Based on Eq. (4-31), the projection from the homogenized stiffnesstensor C to C⁰ can be expressed as

C ⁰ =C ^(iso)=(J::C)J+1/5(K::C)K.  (4-32)

Meanwhile, the Lamé parameters λ⁰ and μ⁰ of the reference material canalso be determined from the isotropic projection. Since C⁰ isapproximated based on C directly, it is guaranteed to be positivedefinite the condition of C. Overall, the projection-based scheme can beconsidered a relaxation of the regression-based scheme.

Summary of the Online Stage

In both schemes, the optimum reference material must be determinediteratively since the values of C in Eq. (4-26) are obtained by solvingthe reduced system with a previous C⁰. A fix-point method is employedhere. For this method, the convergence of the reference materialparameters can be reached in only a few iterations in our experience(i.e., less than five reaches a tolerance of 0.001).

The general algorithm for solving the self-consistent Lippmann-Schwingerequation is as follows:

-   -   1. Initial conditions and initialization: set (λ⁰,μ⁰);        {ε}₀=0;n=0; {Δε}_(new)=0;    -   2. For loading increment n+1, update the coefficients in the        interaction tensor D^(IJ) and the stiffness tensor of the        reference material C⁰;    -   3. Start Newton iterations:        -   (a) compute the stress difference {Δσ}_(new) based on the            local constitutive law (b) use that compute the residual of            the discretized integral equation (4-17):            {r}=ƒ({Δε}_(new),{Δσ}_(new));        -   (c) compute the system Jacobian {M};        -   (d) solve the linear equation {dε}=−{M}⁻¹{r};        -   (e) {Δε}_(new)←{Δε}_(new)+{dε}; and        -   (f) check error criterion; if not met, go to 3(a);    -   4. Calculate (λ⁰,μ⁰) using the regression-based scheme (4-22) or        the projection-based scheme (4-32);    -   5. Check error criterion; if not met, go to step 2;    -   6. Update the incremental strain and stress:        {Δε}_(n+1)={Δε}_(new), {Δσ}_(n+1)={Δσ}_(new);        -   Update the index of loading increment n←n+1; and    -   7. If simulation not complete, go to step 2.

The relative iterative error criterion to the L2 norm of the residual isused. If all the phases of the material are linear elastic, the Newtoniteration will converge in one step. Note that if the self-consistentscheme is not utilized for in the calculation, a constant stiffnesstensor C⁰ will be used, which can be chosen to be same as the matrixmaterial. In this case, C⁰ is not updated, which implies that theinteraction tensors D^(IJ) do not need to be updated, and steps 4-5 inthe algorithm can also be skipped. Although the algorithm with aconstant C⁰ can save time in terms of finding the optimum C⁰, theaccuracy in predict nonlinear material behavior cannot be guaranteedwith a small number of clusters.

Crystal Plasticity Model

In this work, we present an elasto-plastic, anisotropic, heterogeneousplasticity model of the mechanical response of crystalline materials, tobe solved in the SCA framework described in above. The mechanical modelis an implementation of so-called crystal plasticity (CP) constitutivelaws. The finite element scheme used therein to solve the variationalform of the equilibrium equations is replaced with the SCA scheme andits FFT-basis. Thus, in some regards this begins to resemble recentCP-FFT formulations, with the addition of an the offline/onlinedecomposition outlined above.

Brief Overview

Crystal plasticity in conjunction with the finite element method (termedCPFEM) has been applied to solve both microscopic and macroscopicproblems, following from the early combinations of classical plasticityand the finite element method. It has two primary variants: polycrystaland single crystal plasticity. In the polycrystal formulation, eachmaterial point is assumed to represent a collection of crystals suchthat the overall response of the point is homogeneous. In single crystalplasticity, each material point is assumed to represent a singlecrystal, or a point in a single crystal, the deformation of which isgoverned by the particularities of single crystal deformation mechanics(e.g., active slip systems and/or dislocation motion). The formerapproach is more commonly used for macroscopic problems, where arelatively large solution volume is desired. The later is the focus ofthis study. There are many versions of crystal plasticity laws in bothforms. Here the basic kinematics and constitutive law are discussedbelow.

Kinematics

The deformation gradient F can be multiplicatively decomposed as:

F=F ^(e) F ^(p)  (4-33)

where the plastic part F^(p) maps points in the reference configurationonto an intermediate configuration which is then mapped to a currentconfiguration through the elastic part F^(e). Note that physically F^(p)is associated with the dislocation motion and F^(e) is a combination ofthe elastic stretch and rigid body rotation.

The effect of dislocation motion is modeled by relating the plasticvelocity gradient {tilde over (L)}^(p) in the intermediate configuration(usually denoted by {tilde over (□)}) to simple shear deformationγ^((α)):

{tilde over (L)} ^(p)=Σ_(α=1) ^(N) ^(slip) {dot over (γ)}^((α))({tildeover (s)} ^((α)) ⊗ñ ^((α)))  (4-34)

where ⊗ is the dyadic product, N_(slip) is the number of slip systems,{dot over (γ)}^((α)) is a shear rate, {tilde over (s)}^((α)) is the slipdirection, and ñ^((α)) is the slip plane normal, all for a crystal slipsystems (α) in the intermediate configuration. The relationship between{tilde over (L)}^(p) and F^(p) is given by

{tilde over (L)} ^(p) ={dot over (F)} ^(p)·(F ^(p))⁻¹.  (4-35)

Constitutive Law

The final task in constructing the crystal plasticity framework isdefining the constitutive laws of elasto-plasticity. We choose a basisof the Green-Lagrange strain E^(e) and Second Piola-Kirchhoff stressS^(e), from the many conjugate pairs available, which are related by:

S ^(e) ={tilde over (C)}·E ^(e)=1/2{tilde over (C)}·[(F ^(e))^(T) F ^(e)−I],  (4-36)

where the elastic stiffness tensor {tilde over (C)} is defined in theintermediate configuration.

A phenomenological power law for the plastic shear rate in each slipsystem given by

$\begin{matrix}{{\overset{.}{\gamma}}^{(\alpha)} = {{\overset{.}{\gamma}}_{0}{\frac{\tau^{(\alpha)} - a^{(\alpha)}}{\tau_{0}^{(\alpha)}}}^{({m - 1})}\left( \frac{\tau^{(\alpha)} - a^{(\alpha)}}{\tau_{0}^{(\alpha)}} \right)}} & \left( {4\text{-}37} \right)\end{matrix}$

is used, where τ^((α)) is the resolved shear stress, a^((α)) is abackstress that describes kinematic hardening, {dot over (γ)}₀ is areference shear rate, τ₀ ^((α)) is a reference shear stress thataccounts for isotropic hardening, and m is the material strain ratesensitivity. Shear stress is resolved onto the slip directions with:

τ^((α))=σ:(s ^((α)) ⊗n ^((α))),  (4-38)

where σ, s^((α)) and n^((α)) are the Cauchy stress, slip direction andslip plane normal respectively, all of which are in the currentconfiguration. The Cauchy stress is given by:

$\begin{matrix}{{\sigma = {\frac{1}{J_{e}}\left\lbrack {F^{e} \cdot S^{e} \cdot \left( F^{e} \right)^{T}} \right\rbrack}},} & \left( {4\text{-}39} \right)\end{matrix}$

where J_(e) is the determinate of F^(e). The relationship betweens^((α)) and {tilde over (s)}^((α)) is given by

s ^((α)) =F ^(e) ·{tilde over (s)} ^((α)),  (4-40)

and the relationship between n^((α)) and ñ^((α)) is given by

n ^((α)) =ñ ^((α))·(F ^(e))⁻¹  (4-41)

which ensures that the slip plane normal vector remains orthogonal tothe slip direction in the current configuration.

The reference shear stress τ₀ ^((α)) evolves based on the expression:

{dot over (τ)}₀ ^((α)) =HΣ _(β=1) ^(N) ^(slip) q ^(αβ){dot over(γ)}^((β)) −Rτ ₀ ^((α))Σ_(β=1) ^(N) ^(slip) |{dot over (γ)}^((β))|,  (4-42)

where H is a direct hardening coefficient and R is a dynamic recoverycoefficient and q^(αβ) is the latent hardening ratio given by:

q ^(αβ)=χ+(1−χ)δ_(αβ)  (4-43)

where χ is a latent hardening parameter. The backstress a^((α)) evolvesbased on the expression:

{dot over (a)} ^((α)) =h{dot over (γ)} ^((α)) −ra|{dot over (γ)}^((α))|,  (4-44)

where h and r are direct and dynamic hardening factors respectively.

A computational crystal plasticity algorithm needs to solve a set ofnon-linear equations from Eq. (4-33) to Eq. (4-44). Different numericalmethods can be used to solve these equations. McGinty and McDowell gavean implicit time integration algorithm for the material law with thefinite element method. However, the SCA method uses Fast FourierTransformation method, CP alorithms have been shown to be effective inthis framework as well. Here we simply implement the same crystalplasticity model in our SCA and FEM calculations, albeit with a slightvariation in how the deformation gradient, F, is computed.

EXAMPLES

In this section, three example cases probing the capabilities of SCA areimplemented with the CP routine described. First, the SCA method isvalidated for a multi-inclusion system with J2 plasticity. Second, asimple case of a spherical inclusion in a single-crystal matrix isshown. Finally, the complexity of the system is increased by simulatinga polycrystalline cube with equiaxed, randomly oriented grains.

Multi-Inclusion System with J2 Plasticity

The SCA method is firstly validated for a multi-inclusion system using amuch simpler material law: J2 plasticity. The inclusion phase is elasticwith Young's modulus E_(i)=500 MPa and Poisson's ratio v_(i)=0.19. Thematrix phase is elasto-plastic with E_(m)=100 MPa and v_(m)=0.3 in theelastic regime, and it has a von Mises yield surface (J2 plasticity) anda piece-wise hardening law depending on the effective plastic strainε_(p):

$\begin{matrix}{{\sigma_{Y}\left( ɛ_{p} \right)} = \left( {\begin{matrix}{0.5 + {5ɛ_{p}}} & {ɛ_{p} \in \left\lbrack {0,0.04} \right)} \\{0.7 + {2ɛ_{p}}} & {ɛ_{p} \in \left\lbrack {0.04,\infty} \right)}\end{matrix}{{MPa}.}} \right.} & \left( {4\text{-}45} \right)\end{matrix}$

The inclusions are identical to each other and the volume fraction ofthe inclusion phase is equal to 20%. The mesh size for the high-fidelityfinite element model is 80×80×80. The clustering results based on thestrain concentration tensor A(x) are shown in FIG. 67. The numbers ofclusters in the matrix and inclusion are denoted by k_(m) and k_(i),respectively. Note that A(x) has 36 independent components which need tobe determined by elastic simulations under 6 orthogonal loadings. Sincethe volume fraction of the inclusion phase is 20%, we choosek_(i)=┌k_(m)/4┐, where ┌□┐ denotes the nearest integer greater than orequal to □.

The stress-strain curves predicted by the regression-based andprojection-based self-consistent schemes are given in FIG. 68 foruniaxial tension and pure shear loading conditions, where the solidlines represent the DNS results for comparison. The DNS results areplotted as solid lines for comparison. For both schemes, the predictionsconverge to the DNS results by increasing the number of clusters in thesystem, but the regression-based scheme has a better accuracy than theone based on isotropic projection of the effective stiffness tensor. Theaccuracy of the projection-based scheme can be greatly improved throughweighted projection.

For this 80×80×80 mesh, the DNS based on FEM typically takes 25 hours on24 cores (in a state-of-the art high performance computing cluster withtwo 12-core/processor Intel Haswell E5-2680v32.5 GHz processors percompute node). With the same number of loading increments, the SCAreduced order method (in MATLAB) only takes 0.1 s, 2 s and 50 s on oneIntel i7-3632 processor for k_(m)=1, 16 and 256, respectively.

Spherical Inclusion with Crystal Plasticity

The crystal plasticity law is introduced for the simplest geometric casehere, that approximating the 3D Eshelby problem: a sphericalinclusion/void embedded in an infinite (periodic boundary)single-crystal matrix. A schematic of the geometry is shown in FIG. 69.In the context of AM, provided in the introduction, this could bethought of as a spherical void occurring in the interior of a part. Suchvoids, between 1-2 microns and 50 microns diameter are often attributedto boiling and material vaporization during the build process. To modelthis, an nearly infinitely compressible, very low modulus material lawis applied within the sphere, while the CP model is used for the matrixmaterial. A set of crystal plasticity parameters are listed in Table4-3; the Young's modulus and Possion's ratio of the soft inclusion are500 MPa and 0.19, respectively. These parameters match reasonably wellwith a FCC metal, though are not yet calibrated for AM materialsspecifically.

TABLE 4-1 Crystal plasticity parameters for a FCC metal C₁₁₁₁, MPaC₁₁₂₂, MPa C₂₃₂₃, MPa 40356 20257   {dot over (γ)}₀, s⁻¹ m initial τ₀,MPa .002   10 320  H, MPa R, MPa X   0 1 initial a₀, MPa h, MPa r, MPa.0   500 0

To solve for the overall and local response of this geometry, anappropriate choice of data for the domain decomposition stage must bemade. Using the strain concentration tensor and the elastic responseprovides a reasonable overall match in load history to the DNS solution,but the local solution (near the inclusion) does not match well. Asnoted above, different variables may be used to conduct the clustering.Panel (a) of FIG. 70 shows the elastic DNS solution, and panel (b) ofFIG. 70 shows the clusters built from the strain concentration tensor.Choosing the plastic part of the strain tensor at the onset ofplasticity, contours plotted in panel (c) of FIG. 70, results in thedecomposition shown in FIG. 4d . This gives much higher cluster densitynear the inclusion, and we will show in our future work that this allowsfor much more accurate local solutions. Using the fully developedplastic solution, panel (e) of FIG. 70 gives the clustering shown inpanel (d) of FIG. 70. Uniaxial tension in z direction is applied untilthe whole system has fully yielded. The crystal is orientated with Eulerangles ψ=0°, θ=45°, ϕ=0° (using the Roe convention) with respect to acoordinate system aligned with the global axes.

Once the clusters are determined, total and local solutions for stressand strain can be computed with the SCA reduced order method withcrystal plasticity, denoted as CPSCA, in the online stage. The overallsolution corresponding to the clustering in FIG. 70 is shown in FIG. 71.The first set of solutions match very well in the elastic region, andbegins to develop a slight difference at the onset of plasticity. CPSCAbased on elasticity clustering and onset plasticity clustering givesharder response while that based on fully developed plasticity hassofter response compared to CPFEM results.

For this 40×40×40 mesh, the DNS CPFEM implemented as a user material inAbaqus typically takes 4600 seconds on 24 cores. With the same number ofloading increments, CPSCA (in FORTRAN) only takes 5 seconds on one Inteli7-3632 processor using 16 clusters in the matrix.

Polycrystalline RVE

In this embodiment, CPSCA is used to predict the overall response of aRVE including equiaxed, randomly oriented grains with the fullydeveloped plastic strain tensor calculated a priori as offline database.An example of such a RVE is shown in FIG. 72. FIG. 73 shows thecomparison of overall stress strain curve predicted by CPFEM and CPSCArespectively. FIG. 72 shows RVE including 35 equiaxed, randomly orientedgrains (as shown by the inverse pole figure color map) with 20×20×20 and40×40×40 voxel mesh in panel (a) and (b), respectively. FIG. 73 shows σ₃₃ versus ∈ ₃₃ using CPFEM and CPSCA respectively, showing convergencewith mesh size and number of clusters.

We see that the overall response for the coarser cases converge to verysimilar solutions when element or clusters are added. CPSCA results inharder response than the CPFEM solutions when very coarse clustering(e.g., 1 cluster/grain) is used. This is not an exceptional result,because SCA uses a FFT solution based on the Lippmann-Schwingerequation.

The full 3D solution state for S₃₃ at 5% averaged strain is shown in theopacity and color contour plots shown in FIG. 74, where panel (a) is the203 mesh with CPFEM, panel (b) is the 303 mesh with CPFEM, panel (c) the403 mesh with CPFEM clusters, panel (d) is the 403 mesh with 35clusters, panel (e) is the 403 with 70 clusters, and panel (f) is the403 mesh with 140 clusters. The 35 cluster case has one cluster pergrain, whereas the 140 cluster case has four clusters per grain. Opacityscales with stress level. With this visualization, some differences inthe interior can be observed: in the CPSCA method, stress is generallymore concentrated, and lower outside of the concentration region, whencompared to the CPFEM solutions with more distributed and generallyhigher levels of overall stress. In both solution methods and with allmesh sizes and number of clusters, stress concentrates in grains withhigh Schmid factor. The peak values for the FEM and SCA solutions aregenerally within 10%, while the minimum values differ by more.

Again, the DNS CPFEM implemented as a user material in Abaqus typicallytakes 587 seconds, 5177 seconds, and 31446 seconds for the 20×20×20,30×30×30 and 40×40×40 mesh respectively, on 24 cores. With the samenumber of loading increments, CPSCA (in FORTRAN) only takes 18 seconds,96 seconds and 793 seconds using 1 cluster/grain, 2 clusters/grain and 4clusters/grain respectively on one Intel i7-3632 processor.

In sum, we have presented herein a method to dramatically decrease thecomputational cost associated with conducting microscopic crystalplasticity simulations, of the type that can be used to calibratehomogenization models, or to investigate the mechanics of processespertaining to damage or localization within metals. This was motivatedby a desire to predict the mechanical response of material resultingfrom the additive manufacturing process—a challenge of great interestrecently. In these materials, microscale and mesoscale factors (e.g.,voids of different sizes, grains sizes dependent on physical location)are of interest, necessitating a fast method to predict micromechanicalsolutions over a relatively large volume. The method is demonstratedwith three examples: J₂ plasticity in a multi-inclusion system, a simplevoid-like inclusion embedded within a single-crystal matrix, and apolycrystalline RVE of equiaxed grains.

Example 5 Inverse Modeling Approach for Predicting Filled RubberPerformance

In this example, a computational procedure combining experimental dataand interphase inverse modeling is presented to predict filled rubbercompound properties. The FFT based numerical homogenization scheme isapplied on the high quality filled rubber 3D Transmission ElectronMicroscope (TEM) image to compute its complex shear moduli. The 3D TEMfilled rubber image is then compressed into a material microstructuredatabase using a novel Reduced Order Modeling (ROM) technique, namelySelf-consistent Clustering Analysis (a two-stage offline databasecreation from training and learning, followed by data compression viaunsupervised learning, and online prediction approach), for improvedefficiency and accuracy. An inverse modeling approach is formulated forquantitively computing interphase complex shear moduli in order tounderstand the interphase behaviors. The two-stage SCA and the inversemodeling approach formulated a three-step prediction scheme for studyingfilled rubber, whose loss tangent curve can be computed in agreementwith test data.

Composite materials in general exhibit improved mechanical behaviorscompared to their basic constituents. Such characteristics provide awindow for creating specific materials to satisfy requirements that arehard to meet by materials without specific treatments. It is well-knownthat the properties of Polymer Nano-Composites (PNCs) differ from purepolymer components partly due to an interphase region between polymerand filler. Thus, it is possible to design specific properties by addingfillers into polymer components (such as rubber compounds) to achievedifferent viscoelastic behaviors compared to pure polymers withoutfillers. In this exemplary study, a computational framework for theefficient evaluation of filled rubber properties and interphase propertyinverse modeling for improving filled rubber properties prediction isdisclosed.

In the rubber and tire industry, reduction of loss tangent (or tan(δ))can reduce rolling resistance whereas an increase of loss tangentprovides more friction between the tire and the ground. Experimentalstudies reveal that fillers in polymer compounds indeed result indifferent viscoelastic behavior for PNC vs. pure polymer compound. Forstyrene-butadiene rubber, the addition of carbon black filler reducestan(δ) in the low-temperature region but increases tan(δ) in thehigh-temperature region. Moreover, Brinson et al. conducted a study ofstyrene-butadiene rubber with different fillers and concluded thatfillers enhanced the overall rigidity of the PNC but reduced tan(δ).Therefore, various tire properties can be achieved through customdesigned polymer nano-composite (PNC), or filled rubber.

The cause of change in viscoelastic behavior between PNC and the purepolymer has been studied extensively. Due to the exponential growth ofcomputational power over the past decades, researchers are able toutilize Molecular Dynamics (MD) simulations to capture the effect offillers in polymer compounds by observing the interaction betweenpolymer chains and fillers. Polymer chains aggregate around added fillermaterial, creating a denser layer of polymers. Such results are due tovan der Waals interactions between polymers and fillers. Resultsobtained from MD agree well with experimental observations, where theinterphase exhibit stiffer responses compared to the polymer matrix. Tocharacterize such a change of polymer structure, the interphase can beused to distinguish different viscoelastic behaviors of PNC and purepolymer compound. In this work, a filled polymer system using athree-phase model is considered in the current work: these phases arethe filler, interphase, and polymer. The three-phase model is applied tothe filled rubber sample studied in the present work; this model shouldbe able to capture the difference in viscoelastic performance betweenfilled and unfilled rubber.

Rubber properties can be experimentally measured by dynamic mechanicalanalysis (DMA). The experimental procedures provide viscoelasticproperties, i.e., storage and loss moduli, of the rubber compound. As aresult, DMA provides overall homogenized properties of the rubber.Numerically, macroscopic properties can be determined through varioushomogenization approaches. The aforementioned 3D TEM process has beenapplied to a different filled rubber geometry which has already beenstudied via the Finite Element (FE) method to obtain the local stressresponse. However, in this past work, the filled rubber geometryprovided by 3D TEM as a 3D digital image was only converted to a fineconforming FE mesh which could only be used on a supercomputer. Voxelmeshes of large sizes such as 3D digital images provided by 3D TEM aremore suitable for computational homogenization using the FFT basednumerical method. The digital image can be directly used in the FFTbased algorithm to solve for local and overall responses underdesignated macroscopic boundary conditions. Therefore, the properties ofthe filled rubber can be directly obtained via the FFT algorithm fromusing the 3D image of the filled rubber.

The existing experimental techniques allow measurement of tan(δ) curves(also known as master curves), of filled and unfilled rubber throughDMA. Therefore, if the filled rubber is assumed to be a two-phase modelwith rubber and filler phases, it is possible to conduct a numericalsimulation of its responses at different frequencies using measuredproperties of unfilled rubber and filler. In the present work, theproperty of interphase is unknown, which makes it hard to predict thefilled rubber's master curve through basic constituents: unfilled rubberand filler. In this exemplary example, we combined interphase modelingvia the so-called inverse modeling technique and an efficient FFT-basedhomogenization scheme to compute the master curve of a filled rubberusing a fine mesh reconstructed from 3D TEM. The master curve isvalidated by experimental results to illustrate the efficacy of ourproposed scheme. Comparison of the FFT method and the SCA is performedto examine the improved computational efficiency of this reduced ordermodeling-based approach.

The following sections focus on the experimental study of filled rubberand 3D TEM reconstruction of a filled rubber sample, details on the FFTalgorithm employed, the SCA method, the inverse modeling scheme forinterphase properties computation, and results and discussion. Wedemonstrate an efficient integrated experimental and modeling approachthat combines the merits of an FFT homogenization algorithm, data-drivenSCA, with an inverse modeling technique. With 3D TEM reconstruction of afilled rubber sample and experimental DMA data, we gain an understandingof interphase properties in filled rubber components.

Experimental Study of the Filled Rubber

Preparation of Unfilled and Filled Rubber Samples

The unfilled rubber sample used in the present study is made ofpoly-isoprene (IR2200) with some chemical agents such as sulfur, stearicacid, microcrystalline wax, zinc oxide and etc. In the first stage ofmixing, poly-isoprene and agents other than curing agents were mixed.Then curing agents, such as sulfur, were added. The mixture of polymerand agents was cured with a high temperature to obtain a vulcanizedunfilled rubber sample. The filled rubber sample is made ofpoly-isoprene, silica particles (Zeosil 1165MP), and some chemicalagents. It was obtained in the same way except that silica particles areadded in the first stage of mixing. The formula of these rubber samplesis summarized in Table 5-1.

TABLE 6-1 Formula of rubber samples (weight ratio per hundred rubber)Unfilled rubber Filled rubber sample sample Isoprene (IR2200) 100 100Sulfur (cross-linking agent) 1.5 1.5 Other agents 9.3 9.3 Silica (Zeosil1165MP) None 40

Experimental Results on Master Curve Measurements of Filled and UnfilledRubber Samples

Storage and Loss modulus of filled and unfilled rubber samples weremeasured by a dynamic mechanical analysis (DMA) (TA Instruments, rubberrheometer ARES-G2). Cylindrical samples with a diameter of 8 mm and aheight of 6 mm were used. The measurements were operated at 0.1%oscillatory shear strain in which the material response is in the linearviscoelasticity region. In order to obtain the viscoelastic responses ina wide frequency range, a frequency sweep from 0.5 Hz to 50 Hz wasoperated at a temperature range of −60° C. to 40° C. Master curves wereobtained by time-temperature superposition with the referencetemperature of 25° C. where only horizontal shift was performed. It isclearly indicated in FIG. 75 that the temperature dependence of thehorizontal shift factor aT is well described by theWilliams-Landel-Ferry (WLF) equation. FIG. 76, FIG. 77, and FIG. 78 showthe storage shear modulus, loss shear modulus, and loss tangent of thesesamples respectively.

3D-TEM Reconstruction of Filled Rubber Sample

The filled rubber was made into thin sections with about 100 nmthickness using a focused ion beam (JEOL, JEM-9310FIB) at a cryogenictemperature at −150° C. The ultrathin section was transferred onto a Cumesh grid with a polyvinyl formal supporting membrane. Prior to theelectron microscopy experiments, gold particles 5 nm in diameter wereplaced on the ultrathin section from colloidal aqueous solution.

We conducted 3D observations by TEM and 3D-TEM using a JEM-2200FSmicroscope (JEOL, Ltd., Japan) operated at 200 kV. The electronmicroscope was equipped with a slow-scan USC 4000 CCD camera (Gatan,Inc., USA). Elastically scattered electrons (electron energy loss: 0±40eV) were selected by an energy filter installed in the microscope (Ωfilter, JEOL Ltd., Japan).

A series of TEM images were acquired at tilt angles in the range of −66°to 73° at an angular interval of 1°. Subsequently, the tilt series ofthe TEM images were aligned by the fiducial marker method, using goldnanoparticles as the fiducial markers. The tilt series of TEM imagesafter the alignment were reconstructed by filtered back-projection(FBP). It took us about 2 hours and a few days, respectively, to take140 TEM tilt images on TEM and to align those projections before the FBPreconstruction. In addition, segmenting out the fillers from the rubbermatrix in each digitally-sliced image has been done before stacking themto generate a 3D image, which takes, typically, 1 to 2 weeks. The basicprotocol used here is essentially the same as the one in which wedemonstrated less than 1 nm resolution, i.e., 0.5 to 0.8 nm. Thereconstructed filled rubber is shown in FIG. 79, where 3D-image ofmeasured filler structure is shown in top view and in front view, andrubber matrix material is hidden.

Fast Fourier Transform Homogenization Scheme

Formulation of Fast Fourier Transform Scheme

3D TEM generates a high-fidelity 3D reconstruction of a filled rubbersample as a 3D digital image with a resolution of the sub-nanometer, asmentioned in the previous section. The nature of 3D TEM leads to verylarge structured voxel meshes making it hardly feasible for FEcomputation due to the fine resolution. The FE mesh was generateddirectly from the high-resolution 3D TEM data as a voxel mesh with alarge total number of degrees of freedom. To make the reconstructionsuitable for FE analysis, the 3D digital image had to be converted intoa conforming mesh. However, such process required extra time andresources in preprocessing before the actual simulation. Recentdevelopment in FFT homogenization scheme provides an alternate solutionfor solving boundary value problems using a structured mesh. The FFThomogenization scheme avoids this conversion process and makescomputation straightforward using the 3D digital image from the 3D TEMprocess. Although the convergence of the FFT scheme for arbitrary phasecontrasts and its efficiency can still be improved, it is a powerfulhomogenization scheme for 3D images. For example, the FFT schemebypasses the need for mesh generation, which is required by the FEmethod, and reduces the problem size so that it can be solved on apersonal computer. Although the large input image might require largercomputer memory due to an increase of the total number of degrees offreedom, the FFT scheme enables a much easier approach when ahigh-resolution 3D digital image is provided.

In this exemplary example, we first adopt the FFT scheme assuming smallstrain based on the system of equations shown in Eq. (5-1).

$\begin{matrix}\left\{ \begin{matrix}{{{\nabla.{\sigma(X)}} = 0},{\forall{X \in \Omega_{0}}}} & ({equilibrium}) \\{{\sigma(X)} = {f\left( {X,ɛ} \right)}} & \left( {{constitutive}\mspace{14mu}{law}} \right) \\{ɛ = {{sy{m\left( {\nabla u^{*}} \right)}} + ɛ^{Macro}}} & ({compatibility})\end{matrix} \right. & \left( {5\text{-}1} \right)\end{matrix}$

where the applied displacement field u* is periodic over the computationdomain Ω₀. The equilibrium condition for any input mesh is given asbelow following conservation of linear momentum:

∇·σ(X)=0,∀X∈Ω ₀  (5-2)

In the FFT scheme, each voxel in the input image represents a materialpoint. The location of the voxel is represented by X. Stress and straintensors at all material points are computed in the FFT scheme. The localstress σ(X) can be computed using any given constitutive law, but weassume linear elasticity for now.

To solve Eq. (5-2), the FE approach would formulate the Cauchy momentumequation with periodic boundary conditions. Here, the local strain isgiven as in Eq. (5-1). It is composed of sym(∇u*), the symmetric part ofthe gradient of the periodic displacement field u*, and ε^(Macro), theprescribed macroscopic strain tensor. Introducing the polarizationstress T and the Green's operator Γ⁰, it is possible to express to thelocal strain as a Lippmann-Schwinger equation:

ε(X)=−(

⁰*τ)(X)+ε^(Macro)  (5-3)

The polarization stress τ and the explicit form of the Green's operator

⁰ in Fourier space are defined in Eq. (5-4)

τ(X)=

⁰:ε(X)−σ(X)  (5-4)

where

⁰ is the standard stiffness tensor of an isotropic reference material,written as C_(klmn) ⁰=λ⁰δ_(kl)δ_(mn)2μ⁰ δ_(km)δ_(ln) in index notation,with reference Lamé parameters λ⁰ and μ⁰.

$\begin{matrix}{{{\hat{\Gamma}}_{klmn}^{0}(\xi)} = {\frac{\delta_{km}\xi_{l}\xi_{n}}{2\mu^{0}{\xi }^{2}} - {\frac{\lambda^{0}}{2{\mu^{0}\left( {\lambda^{0} + {2\mu^{0}}} \right)}}\frac{\xi_{k}\xi_{l}\xi_{m}\xi_{n}}{{\xi }^{4}}}}} & \left( {5\text{-}5} \right)\end{matrix}$

where the indices of {circumflex over (Γ)}_(klmn) ⁰ coincide with thoseof C_(klmn) ⁰.

Since the explicit form of the Green's operator is known only in Fourierspace, the convolution term in Eq. (5-4) is computed with the help ofthe inverse Fourier transform as in Eq. (5-6).

⁰*τ(X)=

⁻¹{{circumflex over (Γ)}_(klmn) ⁰(ξ)

[τ_(kl)(X)]}  (5-6)

where

and

denote respectively the FFT and the inverse FFT.

To solve the above FFT formulation, various iterative methods can beused, such as fixed point iteration and conjugate gradient. The solutiontechniques are vastly available in the literature. For demonstrationpurposes, the FFT algorithm is presented using fixed point iteration inAppendix A.

The iteration process of the FFT algorithm starts from a given initiallocal strain and checks for convergence on the local ε(X). During theiteration process, Green's function enforces the compatibility conditiongiven in Eq. (5-1). To obtain the macroscopic stress and strain tensorsfrom the mesh, volumetric average following Hill's lemma are conductedas in Eq. (5-7):

$\begin{matrix}{\sigma^{Macro} = {{\frac{1}{V}{\int_{V}{{\sigma(X)}{dV}\mspace{14mu}{and}\mspace{14mu} ɛ^{Macro}}}} = {\frac{1}{V}{\int_{V}{{ɛ(X)}dV}}}}} & \left( {5\text{-}7} \right)\end{matrix}$

To obtain effective elastic material properties μ^(Macro) and λ^(Macro),σ^(Macro) and ε^(Macro) are plugged into Hooke's law as in Eq. (5-8):

σ_(ij) ^(Macro)=λ^(Macro)ε_(kk) ^(Macro)δ_(ij)+2μ^(Macro)ε_(ij)^(Macro)  (5-8)

It is convenient to compute μ^(Macro) and λ^(Macro) by solving Eq.(5-8). One can re-write Eq. (5-8) in matrix format and solve forμ^(Macro) and λ^(Macro).

Application of FFT Scheme for Frequency Domain Computation

For rubber materials or viscoelastic materials in general, responses aredrastically different for various loading frequencies; DMA is a commonexperimental method to evaluate this variation. DMA providesviscoelastic material properties, such as the complex Young's modulus,E*=E′+iE″, and the complex shear modulus, at different frequency points,denoted as ω_(k). The ratio between the E″, the imaginary part of thecomplex Young's modulus, and E′, the real part of the complex Young'smodulus, gives the tan(δ) curve. Complex young's modulus can beinterchangeable with complex shear modulus G*=G′+iG″.

For DMA, a sinusoidal strain with a given frequency ω_(k) is applied tothe rubber and the steady state stress is measured to computeviscoelastic material properties. In the 1D case, the stress at a givenpeak strain ε₀ ^(Macro) can be written as:

$\begin{matrix}{{\sigma^{Macro}(t)} = {{\Re\left\lbrack {ɛ_{0}^{Macro}{E^{*}\left( \omega_{k} \right)}e^{i\omega_{k}t}} \right\rbrack} = {\Re\left\lbrack {{ɛ_{0}^{Macro}\left( {{E^{\prime}\left( \omega_{k} \right)} + {{iE}^{''}\left( \omega_{k} \right)}} \right)}\left( {{\cos\left( {\omega_{k}t} \right)} + {i{\sin\left( {\omega_{k}t} \right)}}} \right)} \right\rbrack}}} & \left( {5\text{-}9} \right)\end{matrix}$

where, for any complex number z,

[z] is the real part of z.

A steady-state solution for σ^(Macro)(t) can be found by simply taking

${t = 0},\frac{2\pi}{\omega_{k}},\frac{4\pi}{\omega_{k}},\ldots$

The steady state stress can then be treated as a complex one, written asσ^(Macro,*)=ε₀ ^(Macro)(E′(ω_(k))+iE″(ω_(k)). σ^(Macro,*) is composed ofa real part and an imaginary part. For the 1D case, taking the quotientof σ^(Macro,*) and ε₀ ^(Macro) yields the complex Young's modulus.Reciprocally, the complex stress can be computed by the FFT scheme byinputting the complex Young's modulus.

The above operation is to be performed at a given frequency point ω_(k).To compute the rubber's tan(δ) curve, an individual tan(δ) point atdifferent ω_(k) is needed, where tan(δ) is defined as

${\tan(\delta)} = \frac{E^{''}}{E^{\prime}}$

for tensile DMA and

${\tan(\delta)} = \frac{G^{''}}{G^{\prime}}$

for shear DMA. Therefore, by computing the rubber's responses atdifferent frequency points, the complex Young's modulus or shear moduluscan be obtained and tan(δ) can then be computed. When a sufficientnumber ω_(k) is taken, a smooth tan(δ) curve of filled rubber can bereconstructed.

In this exemplary example, shear DMA of both unfilled and filled rubberare conducted to reconstruct master curves. However, due to thelimitation of experimental conditions, only complex shear moduli atdifferent frequencies are available. An assumption is that for a complexshear modulus, a conversion to the Lamé constants is still valid. Thisenables computation of tan(δ) using the FFT scheme at various frequencypoints for the filled rubber by inputting properties of basicconstituents: unfilled rubber and fillers.

A computation using unfilled rubber complex shear moduli at differentfrequency points and filler properties is performed using the meshintroduced above. To make the computation more feasible, the filledrubber domain is shrunk by ½ in all three directions to 513×513×75voxels, where the length of each voxel's edge is 1.62 nm. Poisson'sratio of the unfilled rubber is taken as 0.499 allowing for limitedcompressibility of rubber materials. Note that the aforementionedassumption can be improved by choosing a frequency dependent Poisson'sratio to account for limited compressibility in the glassy state, givenenough experimental data from material characterizations. This will beaddressed in future work. The filler material has Young's modulus E=300MPa and Poisson's ratio ν=0.19. The homogenized complex shear modulusfrom the FFT scheme is used to compute the tan(δ) curve. The computedtan(δ) curve and the experimental result are shown in FIG. 80.

The results in FIG. 80 illustrate the inconsistency between predictionand experimental measurement of tan(δ). It can be deduced from thefigure that before the peak of tan(δ), the two-phase model follows adifferent trend than experimental results. In the low-frequency region(less than 1e5 Hz), tan(δ) is lower than experimental measurements.Between 1e5 Hz and 5e7 Hz, tan(δ) is overpredicted.

The predicted G′ and G″ of filled rubber model are shown in panels(a)-(b) of FIG. 81. In panel (a) of FIG. 81, the prediction of G′ in thelow-frequency region, i.e., between 1e1 Hz and 1e5 Hz, is 50% smallerthan experimental values. In the same region, shown in panel (b) of FIG.81, be predicted loss modulus G″ can be as low as 100% below theexperimental values. Such discrepancy between simulation and predictionresult in lower tan(δ) as shown in FIG. 80. In the high-frequencyregion, between 1e5 Hz to 1e7 Hz, the predicted storage modulus has thesame 50% difference w.r.t. experimental data, but the difference betweenthe predicted G″ and experimental data decreases, resulting in a trendof increasing G″. This explains why the predicted tan(δ) shown in FIG.80 between 1e5 Hz and 1e7 Hz is higher than the experimental data. Inthe region of frequency higher than 1e7 Hz, both predicted G′ and G″ areclose to experimental results, giving a relatively good prediction oftan(δ).

Such an inconsistency observed in FIG. 80 is expected, as the input meshonly considered rubber and filler phases. As mentioned in theintroduction, the existing literature supports the assumption that thereis an interphase region between the filler and pure rubber that shouldbe treated as a third material. Specifically, the interphase should havea larger G″ compared to unfilled rubber in the lower frequency regionbetween 1e1 Hz and 1e5 Hz. It should also have a larger G′ compared tothe unfilled rubber in the frequency region between 1e5 Hz and 1e7 Hz.The original two-phase model is incomplete and cannot fully reveal theproperty of filled rubber, which would be required to model theinterphase based on the filled rubber 3D image reconstructed. Oneproblem that emerges after modeling the interphase in the originalfilled rubber mesh is that the property of the interphase stays unknown.Due to a limitation of experimental techniques, the viscoelasticproperty of interphase, or its complex Young's or shear modulus cannotbe given as a known input. Therefore, we propose an inverse modelingtechnique to compute the interphase property quantitatively, describedin the next section, with the aim of producing a better prediction oftan(δ) of filled rubber. However, the inverse modeling procedurerequires multiple iterations and FFT procedure is performed in alliterations. This will impose a considerable computational cost. Toaddress the computational efficiency issue, a reduced order modelingapproach is also introduced to reduce the computational cost ofevaluating filled rubber responses at each frequency point. The reducedorder modeling approach is then combined with the inverse modelingprocedure to compute interphase properties and filled rubber properties.

In terms of the effect of filler moduli, Appendix C gives predicted G′and G″ of the same filled rubber structure with filler Young's modulifrom 300 MPa (3E+8 Pa) to 3 GPa (3E+9 Pa). The variation of the filledrubber G′ and G″ is very little. This means the predicted filled rubberis insensitive to the filler modulus. However, since the FFT algorithmis sensitive to the contrast between filler and matrix materials, fillerYoung's modulus of 300 MPa is used in the present work.

Efficient Reduced Order Modeling for the Filled Rubber Composite

The aforementioned FFT formulation can compute the effective property offilled rubber, but it requires the computation of all local responses atindividual voxels and thus imposes a high computational cost, thoughmuch less when compared to the FEM. The recently proposed SCA methodprovides an alternative for computing effective properties of arbitrarymicrostructure, such as the filled rubber composite, at reasonable acomputational cost. In this section, the SCA formulation is discussed,providing insight into the physically-based reduced order model.

SCA is a two-stage reduced-order modeling approach. In the offlinestage, two steps are performed: 1) all voxel elements in the mesh areclustered based on arbitrary measurement of similarity in mechanicalresponses, such as strain concentration tensor

; 2) The interaction tensor,

, for each pair of clusters is then computed. The offline stage willgenerate a material microstructure database which contains allinteraction tensors between clusters pairs and volume fraction of eachcluster. After the offline state, the original high fidelity RVE iscompressed into a small number of clusters. In the online stage,discretized Lippmann-Schwinger will compute strain and complex stress ineach clusters and RVE level averaged complex stress and strain at anygiven external loading conditions by solving a boundary value problem.Once the RVE complex stress is identified, tan(δ) will be computedaccordingly. This process is concisely illustrated in FIG. 82. Note thatin the previous section, RVE complex stress at each frequency point iscomputed by FFT. Here, voxels are assumed to be “once responded same,always responds the same”. Therefore, the offline database can beconstructed by once. The database can be used for all frequency pointsafterward for an efficient evaluation of tan(δ) at different frequencypoints.

The Lippmann-Schwinger equation given in Eq. (5-3) can be reformulatedin the following form in Eq. (5-10)

ε^(Macro)−ε(X)−∫_(Ω)

⁰(X,X′):[σ(X′)−

⁰:ε(X′)]dX′=0,X∈Ω  (5-10)

where X is the voxel element location in the mesh and X′ is the locationin the reference domain.

To perform reduced order modeling of the filled rubber composite, onecan reduce overall degrees of freedom in the mesh by grouping voxelswith similar mechanical responses together. This process is also knownas clustering. A convenient criterion chosen here is the well-knownstrain concentration tensor

that connects macroscopic prescribed strain to local strain responses ateach voxel, shown in Eq. (5-11) below:

ε(X)=

(X):ε^(Macro) ,X∈Ω  (5-11)

where

is a 6 by 6 matrix in Voigt notation. Six traction free loadings on theoriginal RVE are needed in order to determine all 36 entries of

. Clustering algorithms, such as k-means clustering, can be used tocluster all voxels and decompose the original domain of 19,737,675voxels into K clusters, where K equals 64. This is the first step of theoffline stage. One might think this as reducing total integration pointsto K, where K is a smaller number comparing to the total number ofintegration points in the original mesh. Note that

is not the sole solution for the clustering process. For differentproblems, one might wish to use other meaningful quantities to identityvoxels with similar mechanical responses, such as lattice orientation.

It is convenient to define a characteristic function as in Eq. (5-12) inorder to decompose Eq. (5-10) to incorporate the newly decomposeddomain.

$\begin{matrix}{{\chi^{I}(X)} = \left\{ \begin{matrix}1 & {X \in \Omega^{I}} \\0 & {otherwise}\end{matrix} \right.} & \left( {5\text{-}12} \right)\end{matrix}$

where I=1,2,3, ,K. The discretized Lippmann-Schwinger equation is givenin Eq. (13) for each cluster.

$\begin{matrix}{{{{ɛ^{Macro} - {\frac{1}{c^{I}{\Omega }}{\int_{\Omega}{{\chi^{I}(X)}{ɛ(X)}{dX}}}} - {\frac{1}{c^{I}{\Omega }}{\int_{\Omega}{\int_{\Omega}{{\chi^{I}(X)}{\Gamma^{0}\left( {X,X^{\prime}} \right)}}}}}}:{\left\lbrack {{\sigma\left( X^{\prime} \right)} - {{\mathbb{C}}^{0}:{ɛ\left( X^{\prime} \right)}}} \right\rbrack{dX}^{\prime}{dX}}} = 0},{X \in \Omega}} & \left( {5\text{-}13} \right)\end{matrix}$

where c^(I) is the volume fraction of cluster I and |Ω| is the totalvolume of the mesh.

By noticing that σ(X′) and ε(X′) can be written as:

$\begin{matrix}{\mspace{79mu}{{{\sigma\left( X^{\prime} \right)} = {\sum\limits_{J = 1}^{K}{{\chi^{I}\left( X^{\prime} \right)}\sigma^{J}}}},{{ɛ\left( X^{\prime} \right)} = {\sum\limits_{J = 1}^{K}{{\chi^{J}\left( X^{\prime} \right)}ɛ^{J}}}}}} & \left( {5\text{-}14} \right) \\{\mspace{79mu}{{Now}\mspace{14mu}{{Eq}.\mspace{14mu}(13)}\mspace{14mu}{{becomes}:}}} & \; \\{{{{ɛ^{Macro} - {\frac{1}{c^{I}{\Omega }}{\int_{\Omega}{{\chi^{I}(X)}{ɛ(X)}{dX}}}} - {\frac{1}{c^{I}{\Omega }}{\sum\limits_{J = 1}^{K}{\int_{\Omega}{\int_{\Omega}{{\chi^{I}\left( X^{\prime} \right)}{\chi^{J}\left( X^{\prime} \right)}{\Gamma^{0}\left( {X,X^{\prime}} \right)}}}}}}}:{\left\lbrack {{\sigma\left( X^{\prime} \right)} - {{\mathbb{C}}^{0}:{ɛ\left( X^{\prime} \right)}}} \right\rbrack{dX}^{\prime}{dX}}} = 0}\ ,{X \in \Omega}} & \left( {5\text{-}15} \right) \\{\mspace{79mu}{{{{where}\mspace{14mu}{\mathbb{D}}^{IJ}\mspace{14mu}{{is}:\mspace{79mu}{\mathbb{D}}^{IJ}}} = {\frac{1}{c^{I}{\Omega }}{\int_{\Omega}{\int_{\Omega}{{\chi^{I}(X)}{\chi^{J}\left( X^{\prime} \right)}{\Gamma^{0}\left( {X,X^{\prime}} \right)}}}}}},{X \in \Omega}}} & \left( {5\text{-}16} \right)\end{matrix}$

After the first step of the offline stage process, which is theclustering process,

^(IJ) can be computed. Once

^(IJ) is computed, the second step of the clustering process iscompleted and the original RVE is compressed into a microstructuraldatabase made of clusters and interaction tensors. Plug in

^(IJ) into Eq. (5-13) will give the final form of the discretizedLippmann-Schwinger equation in Eq. (5-17):

$\begin{matrix}{{{ɛ^{Macro} - ɛ^{I} - {\sum\limits_{J = 1}^{K}{{\mathbb{D}}^{IJ}:\left\lbrack {\sigma^{J} - {{\mathbb{C}}^{0}:ɛ^{J}}} \right\rbrack}}} = 0},{I = 1},2,3,\ldots\mspace{14mu},K} & \left( {5\text{-}17} \right)\end{matrix}$

where the incremental form is given as:

$\begin{matrix}{{{{\Delta ɛ^{Macro}} - {\Delta ɛ^{I}} - {\sum\limits_{J = 1}^{K}{{\mathbb{D}}^{IJ}:\left\lbrack {{\Delta\sigma^{J}} - {{\mathbb{C}}^{0}:{\Delta ɛ^{J}}}} \right\rbrack}}} = 0},{I = 1},2,3,\ldots\mspace{14mu},K} & \left( {5\text{-}18} \right)\end{matrix}$

The online stage involves the evaluation process of Eq. (5-18). Thesolution procedure of Eq. (5-18) is given in Appendix B for readers'reference.

The above SCA formulation is combined with Eq. (5-9) to compute theeffective complex moduli of filled rubber at a reduced computationalcost due to the reduction of the number of integration points anddegrees of freedom. The ROM of the original filled rubber domain withclusters is shown in FIG. 83 for 32 clusters per phase. A second ROMwith 64 clusters per phase (making 128 clusters in total) is alsogenerated but not shown for the sake of space. The corresponding mastercurve is given in FIG. 84, where the comparison with the FFT result isshown. The master curves computed by the ROM show a trend of the overalleffective filled rubber performance that is consistent with the resultscomputed by the FFT algorithm. Although SCA over-predicts tan(δ) curve,the results were obtained with a dramatic reduction of computation timewith a speedup of 924 at 64 clusters (32 clusters in the filler phaseand 32 clusters in the unfilled rubber phase). The comparison betweencomputation time using SCA and FFT are shown in Table. 2 below. In FIG.85, the filled rubber G′ and G″ computed by SCA are in the same trend.However, SCA provides considerable savings in terms of computation time.The difference between FFT and SCA predictions are within one order ofmagnitude, this means the ROM doesn't sacrifice all microstructureinformation after the clustering process. Both FFT and SCA predict thesame trend of filled rubber G′ and G″, meaning that the deviation causedby the interphase needs to be captured as mentioned in the previoussection.

To further investigate interphase properties using our reduced ordermodel approach, the above procedure is integrated into the inversemodeling process to better predict filled rubber properties.

TABLE 5-2 Comparison of Computation Time over 17 Frequency PointsComputation Time FFT 4023 s per frequency point, 68400 s total CPU timeROM (64 clusters) Offline: 7 hr (strain concentration tensor generation)5 hr (clustering + interaction tensor generation) Online: 4.35 s perfrequency point, 74 s total CPU time

Inverse Modeling Scheme for the Interphase

With the master curves shown in FIG. 84, it is clear that a two-phasefilled rubber model is not sufficient in representing the true filledrubber behavior. The cause of this inconsistency between computed tan(δ)curves and experimental measurement can be attributed to the interphaseregion between the filler material and unfilled rubber. Therefore, inthe following discussion, the presence of the interphase between fillerand matrix materials is considered. This should reduce the differencebetween the tan(δ) curves plotted in FIG. 84. The interphase is assumedto surround the filler in a spherical region with a radius β_(IP) of9.74 nm. Note here that the interphase is assumed to be uniform, meaningits thickness is constant throughout the whole domain. A voxel-wisesearch of the filled rubber 3D-image is performed to convert theelements within β_(IP) from the matrix to the interphase.

Inverse Modeling Formulation

The filled rubber model with the interphase can be created following theaforementioned voxel-wise search process, but the viscoelastic behaviorof interphase is still unknown. The interphase is used to suppress theinconsistency between master curves from the FFT homogenization schemeand experimental data, thus its complex Young's or shear moduli atdifferent ω_(k) have to be computed. To predict the unknown interphaseproperty with limited experimental data, a so-called inverse modelingscheme is introduced based on optimization techniques. The objectivefunction of the inverse modeling process can be written as:

$\begin{matrix}{{G^{*{,{IP}}}\left( \omega_{k} \right)} = {{\left( \omega_{k} \right)} = {\min\limits_{{(\omega_{k})} + {(\omega_{k})}}{{norm}\mspace{14mu}\left\lbrack \begin{matrix}{\left( {{G^{\prime,{PMC}}\left( {{\left( \omega_{k} \right)},\omega_{k}} \right)} - {G^{\prime,{EXP}}\left( \omega_{k} \right)}} \right) +} \\{i\left( {{G^{'',{PMC}}\left( {{\left( \omega_{k} \right)},\omega_{k}} \right)} - {G^{'',{EXP}}\left( \omega_{k} \right)}} \right)}\end{matrix} \right\rbrack}}}} & \left( {5\text{-}19} \right)\end{matrix}$

Above goal function states for each given ω_(k), the solution ofinterphase complex shear modulus G*′^(IP)(ω_(k)) is found when thedifference of predicted complex shear modulus G*′^(PMC)(G*′^(IP), ω_(k))and G*′^(EXP)(ω_(k)) is minimized. When ω_(k) is fixed, it is possibleto define a function for the root-finding process as:

f(G*′ ^(IP))=G*′ ^(PMC)(G*′ ^(IP))−G*′ ^(EXP)  (5-20)

where ω_(k) is omitted compared to Eq. (5-10) since the solution isfound for each ω_(k) of interest.

To solve for Eq. (5-20), an iterative method is used to find thesolution of G*′^(IP)(ω_(k)). The derivative of Eq. (5-20) can beformulated as in Eq. (5-21) in order to apply Newton's iterative method:

$\begin{matrix}{{f^{\prime}\left( G^{*{,{IP}}} \right)} = \frac{\left( {G^{*{,{PMC},^{n}}} - G^{*{,{EXP}}}} \right) - \left( {G^{*PMC^{n - 1}} - G^{*{,{EXP}}}} \right)}{G^{*{,{IP}^{n}}} - G^{*{,{IP}^{n - 1}}}}} & \left( {5\text{-}21} \right)\end{matrix}$

With Eq. (5-20) and Eq. (5-21), it is possible to write the iterativeprocess as Eq. (5-22):

$\begin{matrix}{G^{*{,{IP}^{n + 1}}} = {G^{*{,{IP}^{n}}} - \frac{G^{*{,{PMC}^{n}}} - G^{*{,{EXP}}}}{\left\lbrack \frac{\left( {G^{*{,{PMC}^{n}}} - G^{*{,{EXP}}}} \right) - \left( {G^{*{,{PMC}^{n - 1}}} - G^{*{,{EXP}}}} \right)}{G^{*{,{IP}^{n}}} - G^{*{,{IP}^{n - 1}}}} \right\rbrack}}} & \left( {5\text{-}22} \right)\end{matrix}$

where the superscript n denotes the current iteration number and n+1denotes the next iteration. The initial guess for interphase propertiesis set to be the same as an unfilled rubber.

Inverse Modeling for the Filled Rubber Model with Interphase

The proposed inverse modeling method is applied to the aforementionedfilled rubber domain to re-compute the filled rubber master curve. Themesh is the one used above, but with the added interphase. Interphase ofthickness β^(IP)=9.74 nm, which is equivalent to 6 voxels, is added tothe domain to create the filled rubber model with the interphase. Theupdated filled rubber mesh with the interphase is shown on the left ofFIG. 86 below. The corresponding ROM of the above three-phase model hasalso been constructed and shown on the right of FIG. 167. FIG. 86 showsthe ROM with 32 cluster in each phase, meaning the whole filled rubbermesh is compressed into 96 clusters.

For the inverse modeling process, 17 frequency points are picked overthe span of the entire master curve of filled rubber. More points can beused for the inverse modeling process but would increase computationalcost. The experimental procedure of measuring the filled and unfilledrubber master curves have been reported above. Material properties forunfilled rubber and filler materials are also introduced above. Theinverse modeling process is combined with the SCA online prediction tobe the third step of the present scheme. At this point, the three-stepprediction scheme for filled rubber is presented. The results of theinverse modeling will be presented and discussed in the next section.

Result and Discussion on Inverse Modeling Results

Through inverse modeling, a more accurate prediction of the filledrubber master curve is shown in FIG. 87. Predicted tan(δ) curves by boththe FFT and SCA method are in a good match with measured experimentaldata. The SCA result is consistent with the FFT result, meaning thecurrent ROM provides sufficient accuracy in predicting overall behaviorof the filled rubber.

The comparison between the predicted G′ and G″ of the filled rubber andexperimental results are shown in FIG. 88 a) and b), respectively. Itcan be concluded that with the consideration of the interphase, theprediction of tan(δ) reaches a good match with the experimental data.

Through inverse modeling, G′ and G″ of interphase are computed as well,as shown in panels (a)-(b) of FIG. 88. As stated above, the interphaseshould have larger G′ and larger G″ compared to unfilled rubberproperties. Only then the filled rubber G′ and G″ can be improved andmatch with experimental results. The inverse modeled interphaseproperties ensure the prediction of the filled rubber's behavior followsthe experimental data. It is worth noting the interphase G′ and G″predicted by the ROM through SCA is higher than that obtained from FFT,but both SCA and FFT predict the same trend of the filled rubber G′ andG″ as a function of frequency. Such a deviation is expected since the G′and G″ predicted using a 2-phase filled rubber by SCA are smaller thanpredictions made by FFT. Higher interphase G′ and G″ will offset suchdifferences in order to comply with the objective function defined inEq. (5-19).

The difference observed between the filled rubber and unfilled rubbertan(δ) shown in FIG. 87 suggested that the filled rubber would have lessrolling resistance due to smaller tan(δ) measured at a lower frequencyrange (1e4 Hz and 1e6 Hz). It is possible to link such characteristicsto computed interphase properties, in which higher storage and lossmoduli are observed, shown in FIG. 88. This leads to a lower tan(δ) ofthe interphase, leading to a decreased overall hysteresis of the filledrubber by reducing filled rubber tan(δ). Hence, the filled rubber wouldhave less rolling resistance compared to the pure rubber.

On the other hand, FIG. 87 shows little difference between the filledand unfilled rubber tan(δ) in the high-frequency range. This suggeststhat in the high-frequency range both materials should behave in asimilar fashion. The computed interphase storage and loss moduli do notvary much from the unfilled rubber properties as shown above. Therefore,the computed interphase properties are consistent with measured tan(δ)of the filled rubber compound.

Also, the peak tan(δ) of the filled rubber is lower than unfilledrubber. In order to increase tire traction at low temperatures, tan(δ)at lower temperatures should be increased. Based on the computed filledrubber properties, this can be achieved by identifying filler materialthat can form interphase with high loss modulus in the high-frequencyrange. Such a material combination would provide increased damping sothe winter traction can be improved. The present workflow is suitable toinversely model necessary interphase storage and loss moduli that narrowdown the domain for material selection.

In terms of inverse modeling results obtained from SCA, FIG. 87 showshigher interphase G′ and G″ magnitudes by SCA prediction. This is due tothe lower filled rubber G′ and G″ prediction made by SCA as shown inFIG. 85 and the role of the interphase to compensate the differenceobserved between the 2-phase filled rubber structure and the 3-phasefilled rubber structure. Hence, larger magnitudes of interphase G′ andG″ means a stiffer mechanical response of filled rubber, which explainslower tan(δ) observed in FIG. 78. In terms of viscoelastic behavior,both SCA and FFT can capture the glass transition region, as describedby the peak of tan(δ) shown in FIG. 87.

In addition, a dramatic increase in computational efficiency wasobserved for the SCA prediction. The comparison of the computation timeof inverse modeling at a single frequency point is shown in the Table5-3 below, where a 1778 speed-up is achieved by applying the proposedROM compared to FFT. A comparison of computational time in evaluatingfilled rubber properties by different methods is summarized in Table5-4. Even though SCA requires an offline stage computation to generatethe ROM of the filled rubber composite, which is still computationallyexpensive. However, once the database is computed, it can be used forall later evaluation procedure. This provides considerable savings incomputational time in the online stage prediction of effectiveproperties of the filled rubber, as well as the inverse modelingprocess. The SCA method, combined with inverse modeling, opens a newavenue towards material design. Moreover, the same sets of ROM can beused for various material properties to compute filled rubber mechanicalbehaviors at a reasonable cost. It is possible to explore the designspace and get both a decent trend and quantitative description of filledrubber. More importantly, the proposed scheme provides an efficientsolution towards investigating interphase properties of filled rubbermaterials for future design needs even with limited computationalresource.

TABLE 5-3 Comparison of Inverse Modeling CPU Time at Single FrequencyPoint Computational CPU Time FFT about 35,576 s ROM (96 20 s clusters)

TABLE 5-4 Speed Comparison of Filled Rubber Property Evaluation for amodel of size 513 × 513 × 75 using Different Methods Degrees of FreedomMemory Method (count) Required Computation time FEM 200 million 1 TB 1million hr. CPU time (with 6144 cores on a supercomputer) FFT 118million 53 GB 9.88 hr. per frequency point, 168 hr CPU time (5,952speed-up) SCA (96 Offline: Offline: Offline: 19 hr clusters) 118 million53 GB 11 hr (strain concentration tensor Online: Online: generation) 576less than 8 hr (clustering + interaction 1 GB tensor generation) OnlinePrediction: 20 s per frequency point, 340 s CPU time (52,631 speed-up)

Despite the encouraging results observed using the inverse modelingscheme, necessary assumptions were made for this process to be possible.For example, the interphase thickness β^(IP) was assumed to be in acircular region around the filler material. However, it is possible forthe interphase thickness to be a function of the filler curvature orfiller size since the degree of polymer chain aggregation can beaffected by such parameters. The scheme will be extended to includevarying interphase thickness around filler to consider geometricaleffect.

In this exemplary study, an inverse modeling scheme is introduced andillustrated as an effort of quantitatively analyzing the interphaseproperties of a filled rubber compound using high fidelityreconstruction of the filled rubber sample. The FFT scheme enablesefficient computation when the fine 3D digital image is used as input.The test data of unfilled and filled rubber provide enough inputs tosolve an inverse modeling process for interphase properties at eachfrequency point. In addition, SCA, a reduced order modeling scheme, iscombined with the inverse modeling procedure to compute interphaseproperties for the first time. Once the offline stage database isconstructed, the database can be conveniently used at all frequencypoints to compute the whole filled rubber master curve. This novelreduced order modeling approach provides considerable savings in thecomputational cost. The consolidation of SCA and the inverse modelingscheme is an efficient and valuable filled rubber design tool. Thepresent method is general enough and can incorporate other details ofthe microstructure, such as variation of interphase thickness. Theobtained interphase property can enable forward computation of athree-phase filled rubber model in the time domain analysis, such astensile testing. It is believed that the effect of interphase can yieldbetter predictions of filled rubber responses under various loadingconditions, and it shall be addressed in future work.

FFT Scheme Algorithm Flow Chart

The algorithm flow chart for the FFT scheme with fixed point iterationis concisely given as below. The convergence test is used to determineif the local strain ε^(i+1) reached a stable value or not. Theimplementation can be easily done in any programming language, providedthat FFT and inverse FFT packages are readily available.

Initialization:

ε⁰(X)=ε^(Macro) ∀X∈V;

σ⁰(X)=

(X):ε⁰(X),∀X∈V;

Iterate i+1 with ε^(i) and σ^(i) known;

-   -   a)={circumflex over (σ)}^(i)        (σ^(i));    -   b) Convergence test;

$\begin{matrix}\left\{ \begin{matrix}{{{{\hat{ɛ}}^{i + 1}(\xi)} = {{{{\hat{ɛ}}^{i}(\xi)} - {{\hat{\Gamma}}^{0}(\xi)}}:{\hat{\sigma}}^{i}}},{{\forall{\xi \neq 0}};}} \\{{{\hat{ɛ}}^{i + 1}(0)} = ɛ^{Macro}}\end{matrix} \right. & \left. c \right) \\{{ɛ^{i + 1} = {\mathcal{F}^{- 1}\left( {\hat{ɛ}}^{i + 1} \right)}};} & \left. d \right) \\{{{\sigma^{i + 1}(X)} = {{{\mathbb{C}}(X)}:{ɛ^{i + 1}(x)}}};} & \left. e \right)\end{matrix}$

Above algorithm flow chart is for single loading step. Readers caneasily modify it to multiple loading steps by defining multipleε^(Macro) for multiple loading steps.

Self-Consistent Clustering Online Analysis Solution Procedure

SCA requires solving the discretized Lipmann-Schwinger equation based onexternal loading condition, either the fixed strain increment Δε^(Macro)or the fixed stress increment Δε^(Macro). The discretizedLipmann-Schwinger equation is shown as in

$\begin{matrix}{{{{\Delta ɛ^{Macro}} - {\Delta ɛ^{I}} - {\sum\limits_{J = 1}^{K}{{\mathbb{D}}^{IJ}:\left\lbrack {{\Delta\sigma^{J}} - {{\mathbb{C}}^{0}:{\Delta ɛ^{J}}}} \right\rbrack}}} = 0},{I = 1},2,3,\ldots\mspace{14mu},K} & \left( {5\text{-}23} \right)\end{matrix}$

The solution to Eq. (5-23) would be strain tensor ε^(I) in each cluster.In order to use Newton's Raphson method to find a solution to Eq.(5-23), the residual form is given as in Eq. (5-24) below

$\begin{matrix}{{r^{I} = {{{- \Delta}ɛ^{Macro}} + {\Delta ɛ^{I}} + {\sum\limits_{J = 1}^{K}{{\mathbb{D}}^{IJ}:\left\lbrack {{\Delta\sigma^{J}} - {{\mathbb{C}}^{0}:{\Delta ɛ^{J}}}} \right\rbrack}}}},{I = 1},2,3,\ldots\mspace{14mu},K} & \left( {5\text{-}24} \right)\end{matrix}$

For macro strain boundary condition, the residual of macroscopic strainis written as

$\begin{matrix}{{r^{K + 1} = {{{- \Delta}ɛ^{Macro}} + {\sum\limits_{I = 1}^{K}{c^{I}\Delta ɛ^{I}}}}},{I = 1},2,3,\ldots\mspace{14mu},K} & \left( {5\text{-}25} \right)\end{matrix}$

For macro stress boundary condition, the residual becomes

$\begin{matrix}{{r^{K + 1} = {{{- \Delta}\sigma^{Macro}} + {\sum\limits_{I = 1}^{K}{c^{I}\Delta\sigma^{I}}}}},{I = 1},2,3,\ldots\mspace{14mu},K} & \left( {5\text{-}26} \right)\end{matrix}$

Solving for Δε^(I) by minimizing residual r^(I). Linearizing r^(I) withrespect to Δεyields

$\begin{matrix}{{0 = {r^{I} + {\frac{\partial r^{I}}{{\partial\Delta}ɛ^{I}}\delta ɛ^{I}}}},{I = {J = 1}},2,3,\ldots\mspace{14mu},k,{k + 1}} & \left( {5\text{-}27} \right)\end{matrix}$

where Jacobian Matrix

${\mathbb{M}}^{IJ} = \frac{\partial r^{I}}{{\partial\Delta}ɛ^{J}}$

is

^(IJ)=δ_(IJ)

+

:(

−

⁰),I=J=1,2,3, . . . ,k  (5-28)

For macroscopic strain boundary condition, one has:

^(I(k+1))=−

,

^((k+1)I) =c ^(I)

, and

^((k+1)(k+1))=0,I=1,2,3, . . . ,k  (5-29)

For macroscopic stress boundary condition, one has:

^(I(k+1))=−

,

^((k+1)I) =c ^(I)

, and

^((k+1)(k+1))=0,I=1,2,3, . . . ,k  (5-30)

Solving Eq. (5-27) gives 6c¹ that updates all local strain incrementΔε^(I). This process should be repeated until residuals in all clustersare minimized.

Example 6 Image-Based Multiscale Modeling System for MechanicalPerformance of Metal Additive Manufacturing

The common assumptions and methods used for multiscale ormicrostructure-sensitive modeling of materials are generally notappropriate for capturing the performance of additively manufactured(AM) metal. Current approaches often rely upon a RVE or some other formof representative structure (e.g., a representative unit cell or simpleperiodic structure); prediction of response with these structures mightalso be predicated upon simplifying assumptions, such as idealization ofthe microstructure (e.g., as an ellipsoid), periodicity, or statistical(spatial) uniformity. Models that predict minimum performance or captureworst-case scenarios often struggle to make predictions that arequantitatively useful in process design—one of the main goals ofstructure-properties-performance modeling. These difficulties relate toan underlying challenge with AM: the localized process produces a mix ofprocess-dependent and random microstructures, which do not follow eitherroughly deterministic (where worse-case analysis might be useful) orfully stochastic (where statistical uniformity would apply) patterns.The material that results from metal AM is heterogeneous, non-uniform,and highly variable.

In this example, a new paradigm is introduced in mechanical modeling ofmicrostructure dependent failure wherein a fast reduced order approachis applied directly to experimentally imaged microstructures whichpopulate a macroscale component; the behavior of these microstructuresis used to predict the macroscale performance. Knowledge of the processhistory (either from modeling or experiments) can be used to selectwhich microstrcture occurs a each material point in the component-levelmodel. Thus spatial, build-to-build, and part-to-part variations can becaptured. Specifically, this concept will be demonstrated with x-raycomputed tomography images of voids constituting a database of hundredsof thousands of possible microstructures. At each material point in acomponent, the specific microstructure is chosen based on the results ofa process model for AM. Finally, the load history during the component'sexpected service life is predicted, and used to estimate the fatiguelife (or another performance indicator).

By using a database of microstructures and conducting a sampling study(viz Monte Carlo analysis), the need to define a representative volumeelement is alleviated, albeit at a computational expense. We can alsoenable prediction of performance variability and distribution byintroducing some randomness into the selection of each microstructure.The choice of microstrcture at each material point in the component isbased on processing history, making it possible capture the differencebetween, e.g., different toolpaths or choice of processing parameters.In the effect, this appears something like a digital twin, where a givenbuild might be tracked through to a failure profile. One mightalternatively think of this a “virtual experiment,” as each “specimen”tested computationally provides one data point in terms of performance(e.g fatigue life), much like a physical experiment.

The resulting framework has several unique features: location- andhistory-dependent properties and performance prediction, scalability tocomponents or even systems (based on the macro-scale solution methodemployed), and ability to predict variability, in addition tomean/min/max, in behavior throughout the domain

This comes at a cost, naturally. A sufficiently rich database of imagesof microstructures ought to be used; “sufficiently rich” is ambiguousand depends on many factors. Some way to connect processing history to asuitably measure of microstructure that is both location/processingdependent, predictable, reliable, and relevant to the mechanicalperformance is also optimal.

Metal AM for use in structural applications where fatigue loading mightoccur is an excellent challenge with which to demonstrate this method.In AM, relatively unique, dispersed, heterogeneous microstructuralfeatures arise that depend upon a host of factors related to theconditions under which a part is fabricated. A truly predictive modelcan provide confidence in the robustness of designs and a quantifiablesafety assessment, while minimizing the number of experiments required.However, this is only possible when using a model with sufficientdescriptive capability.

Understanding the mechanical properties of AM metals has advancedrapidly in the past few years. Many of these advances have beenexperimental, and predictions of the properties of AM materials such asTi-6Al-4V, SS316L, and Ni-based superalloys have been reported. Onecommonality between these materials is that they may widely varypoint-to-point within a single component, between builds with differentparameters, and between different builds on the same machine with thesame conditions and parameters; the challenges associated with thesevarious sources of variability have been noticed by previouscomputationalists. Experimental efforts have noted this variability,too; for example, Gong et al. and Sheridan et al. show significantvariance between builds with different parameters and within builds withthe same parameters, highlighting the importance of processingparameters, but also process-induced randomness. This kind ofvariability and randomness, particularly where defects are concerned,results in heterogeneous material for which standard prediction methodsmay be ill-suited.

Multiscale modeling is one approach that might be able to captureheterogeneous, non-uniform material responses. For example, Horstemeyerprovides a case of fatigue modeling in heterogeneous materials using ahierarchical multiscale approach based on the multi-stage fatigue modeldeveloped. More recent works have focused on applying models throughoutthe processing and subsequent service life for additive manufacturing torelate processing, structure, and performance. These frameworks arevaluable, and although not multiscale, and provide a starting place forthe current work. Both are relatively deterministic, and critically bothuse process modeling to directly predict microstructure and defects.This implies complete reliance upon the accuracy and veracity of theprocess model to capture all important physics—for defects such as poresan accurate prediction remains an challenge within the AM modelingliterature, particularly at the component scale. When fatigue responsedepends on the precise shape and location of each pore, reliance on amodel might be unwise.

In the following, a sketch of the methods employed to build ourmultiscale model is made, including a thermal model of the AM process,an image- and modeling-based statistical description of voids, and themechanical multiscale model used to predict performance. This isfollowed by a computational example, where prediction of the strain-lifebehavior of a sample of specimens built of Inconel 718 with severaldifferent process parameters is demonstrated.

Methodology

Conceptually, this multi-physics, multiscale method is composed of threemajor parts. The first two are used to generate the required informationthat describes the heterogeneity within the materials, and last usesthis information to make mechanical performance predictions. As such,the first two are what relates the method to AM metals; the third partis the primary contribution and is not necessarily restricted to anyparticular material system.

The first part involves thermal modeling of the build process, so thatthe influence of processing parameters is captured. The second part usessynchrotron x-ray computed tomography images of voids in a prior AMbuild to map, based on the processing information in the first part,possible microstrctures to the component. A Monte Carlo type approach isused to generate many possible instantiations to account for randomfluctuations within the same processing conditions (measured as scatterin the processing-structure relationship). The third part is aconcurrent multiscale stress analysis that uses the instantiations fromthe second part and computational crystal plasticity to estimate thefatigue potency of voids on the micron-scale throughout acentimeter-scale component. A standard test specimen will be used as aconsistent example throughout, although the method is widely applicable.

FIG. 89 shows an overall diagram of the computational scheme. Geometry,build process parameters, material, and loading conditions must bespecified. These are used to conduct a thermal analysis and a macroscalestress analysis. For each material point X within these two models, anelement-wise sub-model is constructed to represent a possible state atthat point. This uses local thermal history and strain history todetermine the microstructure (void geometry) and deformation history.These are used to predict the microscale evolution of state variablessuch as plasticity and damage, which are homogenized (e.g., by takingthe I_(∞) norm of the domain) and used as element-wise estimators ofpart-level susceptibility to failure.

Model Setup

The system described here is applicable to the same geometries, choicesof processing conditions, boundary conditions during mechanical load,and materials that can be represented in standard finite elementsanalysis. Sufficient experimental data, especially 3D images of defectsas will be described later, are required for the material of interest.

Macroscale

Thermal Modeling for the AM Build Process

We start by conducting a thermal analysis to model building thecomponent of interest using the directed energy deposition (DED) method.This analysis is done using a transient thermal Finite Element solver.The governing heat transfer energy balance to be solved is:

$\begin{matrix}{\frac{{\partial\rho}c_{p}}{\partial t} = {{\frac{\partial}{\partial x_{i}}\left( {k\frac{\partial T}{\partial x_{i}}} \right)} + Q}} & \left( {6\text{-}2} \right)\end{matrix}$

where ρ is the material density, c_(p) is the specific heat, t is thetime, x_(i) are the spatial coordinates, k is the conductivity of thematerial, T is the temperature, and Q represents the heat source.

This heat source is represented by a moving laser described by theGaussian distribution:

$\begin{matrix}{Q = {\frac{2\; P\;\eta}{\pi\; R_{b}^{2}}{\exp\left( \frac{{- 2}\left( {x^{2} + y^{2} + z^{2}} \right)}{R_{b}^{2}} \right)}}} & \left( {6\text{-}2} \right)\end{matrix}$

where P is the power of the laser, η is an absorptivity factor to limitthe amount of energy absorbed by the material from the laser which wastaken to be 30%, and R_(b) is the radius of the laser. The variables x,y, and z are local coordinates of the laser. Heat loss on the dynamicfree surfaces of the model is simulated though a combination ofconvection and radiation. Convective heat loss is defined by

q _(conv) =h _(c)(T−T _(∞))  (6-3)

where h_(c) is a convection coefficient, T is the surface temperatures,and T_(∞) is the far-field (ambient) temperature. Radiation heat loss isdefined using the Stefan-Boltzmann law, given by

q _(rad)=σ_(s)ε(T ⁴ −T _(∞))  (6-4)

where σ is the Stefan-Boltzmann constant and ε is the surface emissivityof the material.

Particular build parameters, including laser speed and power and thetoolpath are selected. The material must also be specified. With thisinformation given, the model can predict the time-temperature-history ofeach point within the part. The solidification cooling rate (SCR) iscalculated based upon the temperature history of the thermal model andoutputted at each node. This is approximated, according to Eq. (6-5), asthe time it takes for a material point, represented by subscript i, toreach the solidus temperature from the liquidus temperature. In order tocapture this solidification behavior Eq. (6-1) is solved explicitly withan approximate timestep of 9.0×10⁻⁴ s. If too large of a timestep ischosen, one may skip over the solidification behavior at some materialpoints. Additionally, in the case of re-melting, only the final coolingstage is considered.

$\begin{matrix}{{S\; C\;{R\left( X^{M} \right)}} \approx {\frac{T_{X^{M}}^{liquid} - T_{X^{M}}^{solid}}{t_{X^{M}}^{liquid} - t_{X^{M}}^{solid}}{\forall{X^{M} \in \Omega^{M}}}}} & \left( {6\text{-}5} \right)\end{matrix}$

where SCR(X^(M)) is the solidification cooling rate as a function of themacroscale spatial coordinates X^(M) within the macroscale domain Ω^(M),T_(X) _(M) ^(solid) is the solidus temperature, T_(X) _(M) ^(liquid) isthe liquidus temperature, t_(X) _(M) ^(liquid) is the time at which theliquidus temperature is reached, and t_(X) _(M) ^(solid) is the time atwhich the solidus temperature is reached. However, more information isrequired before a mechanical model of the component can be conducted.Specifically, the anisotropic distribution of microstructure, includingdefects such as voids, developed during the build needs to be estimated.

Microscale

Relate Thermal Conditions and Defects

To use the thermal prediction, we need to connect thermal model outputsto defect statistics and geometry. We choose to generate thisrelationship in two stages: first through a relationship between athermal descriptor (e.g., SCR, as computed above) and microstructuredescriptor statistic, then through a database of microstructuralgeometries which correspond to each microstructural descriptor (e.g.,voids size) statistic. The microstructures from this database is thenused to populate each realization of the part. This defect estimationand database building process is outlined in FIG. 90. In the first part,a relationship between solidification cooling rate (SCR) and void volumefraction (V_(f)) is determined using process modeling and X-ray computedtomography. Panel (a) of FIG. 90 is the subsets of the images acquiredwith X-ray tomography are selected on the basis of V_(f), such that theexpected range of V_(f) for any arbitrary part (with known or predictedthermal history) is spanned. Panel (b) of FIG. 90 is a database of thesepossible microstructures is generated, including computing the trainingstage of the mechanical model.

In order to identify the SCR-to-void relationship and build a database,we used two Inconel DED single-track thin wall parts built withprocessing parameters corresponding to parameter set 1 in Table 6-2.Both walls used a vertical zig-zag toolpath pattern, but while onewall's toolpath was continuous, the other added a one-minute dwellbetween each layer.

X-ray tomography imaging experiments were performed at Beamline 2-BM atthe Advanced Photon Source, Argonne National Laboratory on specimensextracted from 22 locations on the two thin walls (total 44 datapoints). Each image was of about 1 mm³ of material with voxel edgelength 0.65 μm. Contrast from x-ray absorptivity was used to distinguishbetween voids and material using a series of processing steps includingfiltering, thresholding, and artifact removal. Eleven differentdescriptive statistics were extracted from these images, such as voidlocation, size, shape, orientation, and n-nearest neighbor information.For simplicity, we will focus here on the void size, as represented bythe single-point correlation statistic: void volume fraction V_(f). Theoverall V_(f) was computed from the sum of the voids sizes throughoutthe image for each location.

The build process of these two thin walls is modelled using the thermalanalysis method outlined in above. The parts' thermal history issummarized as a point-wise SCR throughout the build. The SCR is auseful, physically relevant single-point statistic that summarizes thethermal conditions that result from the choice of building parameters.The average SCR at the location of each of the x-ray images was computedusing this processing model. Note that experimental measurements of thecooling rate, e.g., with an infrared (IR) camera, could be used toprovide equivalent data. The first part of FIG. 90 shows how the averageSCR versus the V_(f) of each image is gathered and plotted. Anexponential relationship given by:

V _(frac) =Ae ^(−(B)(SCR))  (6-6)

is fit to the data. The fit parameters are A=0.0047 and B=0.0011.

Build Image Database

Next, this relationship is used to determine a range of possiblemicrostructures that might occur within the build. Subsets of the imagesused in the first part were exhumed, such that the V_(f) range of thesubsets corresponds to the range estimated to occur in a part as givenby the relationship shown in FIG. 90. These image subsets constitute adatabase of possible microstructures, in this case each was of size 97.5μm×97.5 μm×97.5 μm and between about V_(f)=0.0001 and V_(f)=0.03.

To complete the database, the three training steps outlined in Sect.2.4.1 were conducted on each entry. Thus, the final database used forthe multiscale mechanical response prediction contains proto-data usedto generate response predictions for arbitrary loading, which dependsupon microstructural information.

Mechanical Response Prediction

In the previous two sections, we have used a database of experimentaldata to populate the part with a realistic, heterogeneous distributionof voids that correspond to the processing conditions used to build thepart. The next step is to use this information to predict the mechanicalproperties of the part. However, for the same reason that it wascomputationally infeasible to directly predict the defect structure, itwould be infeasible to directly model the mechanical response of themicrostructures we have generated.

In order to capture the microstructural information required to predictmechanical performance of AM materials, multiscale approach is taken. Atthe macroscale, a simple Johnson-Cook material model is used to providethe strain boundary conditions (more specifically, deformation gradientsare used) to the data-driven reduced order model used at the microscale.At the microscale, crystal plasticity is used to predict the materialbehavior. Lacking more complete data, we simply assume that anymicrostructure with V_(f) within ±10% of that specified by Eq. (6-6) hasequal probability of occurring. In this way, the part is described byessentially a Monte Carlo process, where the possible states of therandom variable is controlled by the process model. A schematic of thisprocess is shown in FIG. 91, where the local temperature history andstrain history are taken as inputs to a micromechanics model thatreports a homogenized response. For each macroscale material point(element, in this case), the thermal history and strain history arepassed to a microscale solver; a microstructure is selected from thedatabase developed based on the thermal history, and deformationboundary conditions are applied according to the strain history. Acrystal plasticity based microscale solution is computed, and ahomogenized response (e.g., the l_(∞)-norm of the fatigue indicatingparameter, if a fatigue problem is chosen) is returned to themacroscale.

Microscale Reduced Order Model with Crystal Plasticity

At the microscale, the modeling approach combines data-drivenmicromechanics with computational crystal plasticity, termed crystalplasticity self-consistent clustering analysis (CPSCA). The method isderived from first order homogenization, the Hill-Mandel condition, andlocal equilibrium. This problem is defined concisely in a finitedeformation as:

$\begin{matrix}\left( {\begin{matrix}{{\frac{\partial P}{\partial X} = 0},\;{\forall{X \in \Omega}},} \\{{F = \frac{\partial u}{\partial X}},{\forall{X \in \Omega}},} \\{{\frac{1}{\Omega }{\int_{\Omega}^{\;}{F\; d\;\Omega}}} = F^{0}}\end{matrix}.} \right. & \left( {6\text{-}7} \right)\end{matrix}$

where P is the first Piola-Kirkchoff stress, u is the displacement atpoint X in domain Ω, F is the displacement gradient, and F⁰ is theremove (applied) deformation.

To satisfy the Hill-Mandel condition we assume a periodic displacementfield within the microscale domain and anti-periodic boundary traction.Under this assumption, the boundary value problem given in Eq. (6-7) hasbeen shown to be equivalent to the Lippmann-Schwinger equation, andapproximated clusterwise as

F ^(I)+Σ_(J=1) ^(N) ^(c) D ^(IJ):[P ^(J) −C ⁰ : F ^(J)]−F ⁰=0, with I=1,. . . ,N _(c),  (6-8)

where the domain has been discretized into a fixed, finite number ofclusters N_(c), C⁰ is a reference stiffness, and * denotes convolution,and the tensor D^(IJ) describes the interaction between clusters I andJ, given as

$\begin{matrix}{D^{I\; J} = {\frac{1}{\left| \Omega^{I} \right|}{\int_{\Omega}{{\chi^{I}\left( {\Gamma^{0}*\chi^{J}} \right)}d\;{\Omega.}}}}} & \left( {6\text{-}9} \right)\end{matrix}$

where Ω^(I) is the domain in cluster I, Γ⁰ is a periodic fourth orderGreen's operator, and χ is the characteristic function. For a detailedderivation.

CPSCA solves this equation in two stages. The first “training” stageincludes three parts: data collection, data compression (or clustering),and computation of the interaction tensor. The resulting interactiontensor can be stored for future use. The second “prediction” stage makesuse of the interaction tensor and solves the integral equation given inEq. (6-8) subject to an applied strain state and material law.

This first stage is conducted for each of the subset volume, adding theclustering and interaction tensor data to the database of image subsets.

Thus, after a part is instantiated and subsets used to populate themicroscale, only the second stage has to be used compute the mechanicalresponse. The second stage can then be run any number of times, withindependent stress-strain histories and boundary conditions. In thesecond stage, cyclic loading is applied, and the stresses are computedwith a crystal plasticity (CP) material law. The applied deformationgradient is decomposed into an elastic and plastic part; the plasticpart of the deformation gradient is computed from the plastic velocitygradient, which itself is determined by summing the plastic shearvelocity across slip systems in the intermediate configuration, as:

{tilde over (L)} ^(p)=Σ_(α=1) ^(N) ^(slip) {dot over (γ)}^((α))({tildeover (s)} ^((α)) ⊗ñ ^((α)))  (6-10)

where ⊗ is the dyadic product, α is a slip system, N_(slip) is thenumber of slip systems, {dot over (γ)}^((α)) is the microscale shearrate, s^((α)) is the slip direction, and n^((α)) is the slip planenormal. The phenomenological power-law with backstress shown in Eq.(6-11) is used to update the shear slip rate.

$\begin{matrix}{{{\overset{.}{\gamma}}^{(\alpha)} = {{\overset{.}{\gamma}}_{0}{\frac{\tau^{(\alpha)} - a^{(\alpha)}}{\tau_{0}^{(\alpha)}}}^{m}{{sgn}\left( {\tau^{(\alpha)} - a^{(\alpha)}} \right)}}},} & \left( {6\text{-}11} \right)\end{matrix}$

where τ^((α)) is the resolved shear stress (computed withτ^((α))=σ:(s^((α))⊗n^((α))) on slip system α, {dot over (γ)}₀ is areference shear rate, τ₀ ^((α)) is a reference shear stress, a^((α)) isa backstress term, and m is a “rate hardening” coefficient.

Example Performance Measure: Fatigue Life

Failure prediction encompasses a range of damage mechanisms, whichdepend on predictions of properties. One of the more challenging failureto attempt to predict is that caused by cyclic loading: fatigue. Thus,we demonstrate this method with an application to measuring fatigueperformance. Specifically, to predict fatigue crack incubation life, afatigue indicating parameter (FIP) derived from the critical planeapproach is used to estimate the fatigue incubation life of themicrostructure, given a plastic strain history. The FIP is defined by

$\begin{matrix}{{F\; I\; P} = {\frac{\Delta\;\gamma_{\max}^{p}}{2}\left( {1 + {\kappa\frac{\sigma_{n}^{\max}}{\sigma_{y}}}} \right)}} & \left( {6\text{-}12} \right)\end{matrix}$

where Δγ_(max) ^(p) is the maximum cyclic plastic shear strain, σ_(n)^(max) is the peak stress normal to the plane on which Δγ_(max) ^(P)occurs, σ_(y) is the yield stress, and K is a normal stress factorassumed to be 0.55. The FIP is related to number of incubation cyclesusing:

NFIP_(max)=γ _(f)(2N _(inc))^(c)  (6-13)

where γ _(f) and c are multiplicative and exponential Coffin-Manson-likecalibration factors.

Numerical Demonstration

Calibration/Validation Case with Standard Fatigue Specimen

To demonstrate the method, a fatigue specimen conforming to the ASTME606/E606M/E466 standard geometry is numerically tested. The specimengeometry was meshed with two different hexahedral meshes, for onethermal and one stress analysis, as shown in FIG. 92. The two meshes arelargely the same, except the mesh used for stress analysis is coarsenedin the thickness (z-) direction. This is the build direction, and thusrequires at least one element per build layer in the thermal model, butduring the stress analysis the stress and strain are roughly constantthrough this direction. This may, however, smooth out some of thevariability between process parameters, as SCR was averaged during meshcoarsening. Following the schematic shown in FIG. 89, this mesh wasloaded into both the thermal solver then the stress solver. An in-housethermal FEA code was used for the thermal model, C3D8R elements inABAQUS were used to compute stress and strain at the macroscale. Thelower grip was fixed and the upper grip was displaced at constant strainrate over five fully reversed load cycles at several different strainamplitudes.

Macroscale homogeneous material properties corresponding to Inconel 718(IN718) were applied to both the thermal and macroscale stress analyzes.These thermal properties for IN718 are summarized in Table 6-1. TheJohnson-Cook parameters were used, although this choice is essentiallyarbitrary—under the fatigue loading specified, the macroscale responsewas entirely elastic.

TABLE 6-1 Thermo-physical properties of IN718 Property Notation ValueDensity ρ (kg/m³) 8100 Solidus temperature T_(s) (K) 1533 Liquidustemperature T_(l) (K) 1609 Specific heat capacity c_(p) (J/kgK) 360.24 +0.026 T − 4 × 10⁻⁶ T² Thermal conductivity k (W/mK) 0.56 + 2.9 × 10⁻² T− 7 × 10⁻⁶ T² Latent heat of fusion L (kJ/kgK)  272

For the thermal analysis, a zig-zag tool path with 90 layer-by-layeroffset was selected, with the four different process parametersspecified in Table 6-2. The part was meshed and simulated with the gaugesection aligned normal to the build direction. A relatively fine meshwith 539,216 hexahedral elements (panel (b) of FIG. 92, substrate notshown) was used; panel (b) of FIG. 92 shows a detail of the mesh in thenarrowest part of the gauge section. Panel (c) of FIG. 92 shows detailsof the specimen meshes, including the difference between stress (left)and thermal (right) meshes.

TABLE 6-2 Set of process parameters for thermal analysis of IN718Parameter set Laser power Scanning Beam radius Layer (W) speed (mm/s)(mm) thickness (mm) 1800 15 1.5 0.75 1800 10 1.5 0.75 1500 15 1.5 0.751500 10 1.5 0.75

The CP model was used at the microscale. The parameters were calibratedby minimizing the difference between many runs of a cubic domainincluding 64 cubic grains and a set of baseline tensile and cyclicloading data for AM IN718, with starting conditions taken for m and {dotover (γ)}₁ and for elastic moduli. The resulting the model parametersare given in Table 6-3. The parameters that relate FIP to fatigue lifeare fit to experimental high cycle fatigue data of IN718 collected fromliterature. We acknowledge that this generally represents relativelydefect-free material, and may not be perfect for AM material; however,fatigue data for AM IN718 is relatively scant, and producing such datawas outside the scope of this work.

The imaged voids identified by the method above were assumed to beembedded within a single crystal oriented so that the fastest growthdirection was aligned with the build direction. This is consistent withexperimental experience which suggests that grains in IN718 are muchlarger than the voids and preferentially orientated. Future work coulduse method, such as the cellular automata approach to predict grainsthroughout a build. Another possibility would be to syntheticallygenerate grain structures from given experimental evidence whichprovides a statistical basis for the synthetic generation. Preliminary,as-yet-unpublished results indicate that grains generated this way couldsubstantially impact predicted fatigue lives when used in concert withimage-based void geometries.

A snapshot of the thermal response for the part being built under one ofthe processing parameter sets is shown as contours of temperature inFIG. 93, as an example of the thermal prediction. This thermalprediction progresses through the full building process for thisspecimen.

TABLE 6-3 Primary crystal plasticity model parameters Parameter ValueC₁₁, MPa 257,000 C₁₂, MPa 127,000 C₂₃, MPa 94,000 {dot over (γ)}₀, s−10.0024 m 60 τ₀, MPa 360

We simulated thirty different virtual test specimens, each with adifferent, random distribution of defects, and thus estimated fatiguelife. This is shown in FIG. 6, where five realizations of specimens weresimulated at three different load amplitudes for two differentprocessing conditions. Each specimen includes 19,360 elements at themacroscale, each of which is represented by a 150×150×150 voxel mesh atthe microscale. Thus, each specimen is represented, effectively, byabout 65 billion voxels. Put another way, this 30 specimen test suiterequired an evaluation of 580,800 of these microscale volumes, or atotal of 1.9602×10¹² calls of the crystal plasticity routine over 130time steps for each voxel. Without the CPSCA method used at themicroscale, this would be a vast computational expense. However, withCPSCA, each virtual specimen test took about 9 hours using 36 cores inparallel on an Intel Xeon Skylake 6140 at 2.3 GHz clock speed.

FIG. 94 shows estimated fatigue lives for multiple realizations of thefatigue coupon, run at different applied strain amplitudes, mimickingexperimental conditions. Two different processing conditions (conditions1 and 2 in Table 6-2) were modeled. The results in FIG. 94 directlydemonstrate the key features of this method. The contour plots of thegauge sections report an estimated number of cycles required to causefatigue crack initiation at each macroscale point in the fatiguespecimen; at any given macroscale point, the microstructure is differentbetween different instantiations (Specimen A versus Specimen B in FIG.94), which results in different contours plots. Comparing themicrostructures between Specimen A and Specimen B provides the reasonbehind this difference: features with higher fatigue potency might, byrandom chance, occur at the point of highest strain concentration in oneinstantiation but not another. This is similar to behavior seen inphysical testing. Although we assume that voids always occur, this isnot a necessary assumption of the method—given sufficientcharacterization data of the processed material, many classes ofmicrostructure might be seeded using this method.

This microstructures used here do not explicitly capture surface effectsfor the microstructural fatigue behavior. For example, a void on themicroscale near surface of the entire part might have an impact offatigue performance. While the macroscale effects of the boundaries arenaturally included, this small scale interaction is a matter of ongoingwork. Some authors suggest that there is limited change in overallfatigue life, up to the high-cycle limit (runout), for as-built versusmachined finish specimens; this may indicate the assumption made here isreasonable.

A simplifying assumption for the grain structure was also used for thisdemonstration; however, we could add a step that either predicts thegrain structure from thermal history, as was demonstrated, or derives agrain structure from experiments should that data be available.Currently, the kind of 3D grains structures needed to includeimage-based grains are not available to us, although one might considerusing statistically-similar, synthetically generated grain structuresbased on currently available images the next logical step in purelyimage-based microstructures.

An implicit assumption of the images used here is that the track spacingwas appropriate to avoid lack-of-fusion defects between tracks. This isbecause the images used to make the database came from single-track,thin-wall build, were no between-track porosity would be possible. Thisassumption could easily be relaxed by including images of voids inmulti-track builds in the database.

Development of a benchmark for fatigue prediction in AM would supportthese modeling efforts. Currently, conflicting reports of the influenceof AM, versus conventional processing, on the fatigue properties ofmetals exist. This can in part be attributed to the wide range ofmaterials, but also to a range of build processes and choices made bymachine operators. Without a better, AM specific, standard testprocedure such conflicts will likely persist, to the detriment of themodeling community who lack calibration and validation data.

This example presents a method that exploits computational efficientmicromechanics techniques to perform Monte Carlo-style numericalexperiments. The specific microscale solutions are derived from acluster-based solution of the Lippmann-Schwinger equation, and involvethe prediction of fatigue life using crystal plasticity and a fatigueindicating parameter. A database of possible microscale geometries isdeveloped from 3D imaging experiments. These geometries are related toAM processing conditions through the solidification cooling rate, and aprocess- and microstructure dependent, stochastic prediction of fatiguelife is achieved with reasonable computational expense.

Example 7 Efficient Multiscale Modeling for Woven Composites Based onSelf-Consistent Clustering Analysis

In this exemplary embodiment, the curse of computational cost in wovenRVE problem is countered using the SCA, which maintains a considerableaccuracy compared with the standard FEM. The Hill anisotropic yieldsurface is predicted efficiently using the woven SCA, which canaccelerate the microstructure optimization and design of wovencomposites. Moreover, a two-scale FEM×SCA modeling framework is proposedfor woven composites structure. Based on this framework, the complexbehavior of the composite structures in macroscale can be predictedusing microscale properties. Additionally, macroscale and mesoscalephysical fields are captured simultaneously, which are hard, if notimpossible, to observe using experimental methods. This will expeditethe deformation mechanism investigation of composites. A numerical studyis carried out for T-shaped hooking structure under cycle loading toillustrate these advantages.

Woven composites are widely used in industries such as aerospace andautomotive because of their excellent mechanical performances ascompared to unidirectional laminated composites. However, performingstructural analysis of woven composites is challenging due to themesoscale and microscale heterogeneities (see FIG. 95). Differentfeatures can be observed at these different scales, and simplyhomogenizing the composite structure and applying phenomenologicalconstitutive relations that only characterize the average behavior ofthe material does not account the localized behavior at the finerscales. As a result, local nonlinear deformation and damage effects arenot considered. In addition, the macroscale properties cannot bepredicted based on the microstructural constituents, and experiments arerequired to design new composites, which are costly and time consuming.

Multiscale simulation provides a powerful method for analyzing both thematerial microstructure and macrostructure. Using this method allows themacroscale performance of woven composites can be predicted based on theproperties of the constituents. Once the microstructure ischaracterized, macrostructural experiments are not needed every time themicrostructure is changed. This allows the multiscale method toaccelerate material design of woven composites, reduce the cost, andimprove the analysis accuracy of woven composite structures. Moreover,the physical fields in different scales can also be captured, which arehard, or sometimes impossible, to observe using experimental method.Accomplishing effective multiscale simulations for woven compositesstill involves some challenges.

The first challenge is to find an efficient woven RVE solution.Effective macroscale properties are homogenized properties ofcomposites, which are always adopted for the material selection andstructural design with woven composites. To predict these effectiveproperties, an RVE for the woven composite material must be developed,which will establish the link between the microstructural features andeffective macrostructural properties. In the case of a periodic wovenarchitecture, a unit cell is used for the RVE. For the microstructuredesign, the woven RVE solution can be integrated into an optimizationalgorithm in which the RVE has to be solved repeatedly to find theoptimized solution and satisfy the requirement of objective effectiveproperties. Therefore, solving the woven RVE problem efficiently canaccelerate the whole process of optimization. As a result, it willpromote the microstructure design of woven composites. Currently,several approaches have been proposed for solving the RVE problem. Theanalytical approaches, such as mixtures rules and theoreticalmicromechanics methods are efficient, but will lose accuracy in the caseof complex microstructure and nonlinear, history-dependent materiallaws. The Direct Numerical Simulation (DNS) method, such as FEM, isextremely time consuming. The FFT-based method is more efficient thanFEM, but encounters convergence problems for the high phase contrast innonlinear problems. The Transformation Field Analysis (TFA), theNonuniform Transformation Field Analysis (NTFA) and Proper OrthogonalDecomposition (POD) are other solution methods, but they requireextensive a priori simulations to obtain deformation modes, especiallyfor nonlinear phase behavior.

The second challenge is concurrent multiscale simulation for wovenstructures. The behavior of woven composite structures is predictedusing the behavior of the RVEs through the concurrent multiscalesimulation. Additionally, the physical fields in different involvedscales can be captured simultaneously, which will expedite thedeformation mechanism investigation of woven composite structures.Concurrent simulation requires numerous RVE solutions, which iscomputationally expensive using the FE² framework, as shown in FIG. 96.In this example, only 5000 elements are used at the macroscale level,1,843,200 elements are used at the woven RVE mesoscale level, and576,000 elements are used at the UD RVE microstructural level. For theconcurrent multiscale simulation, every up-scale material point will belinked with a down-scale RVE. In this example, assuming a singleintegration point for each element, 2,511.4 trillion elements arerequired for the entire multiscale computation. This computation isextremely expensive, and would require the use of a High-PerformanceComputing Cluster (HPCC).

Solving these challenges require improving the solution efficiency ofthe RVE problem. The SCA is an effective and efficient method to solvethe RVE problem, which can be used for complex woven architectureundergoing irreversible processes, such as inelastic deformation. Thismakes it particularly attractive for integration into a multiscalesimulation. The SCA method involves a two-stage process, an offlinestage and an online stage. In the offline stage, a clustering algorithmis used to reduce the overall degrees of freedom (DOF) of the RVE,resulting in a reduced order RVE. In the online stage, the reduced orderRVE is utilized for solving the discrete incremental Lippmann-Schwingerintegral equation to obtain the stress and strain fields in the reducedorder RVE. This efficient method has been used for simulation for2-dimensional (2D), two-phase composites, and 3-dimensional (3D), hardinclusion material considering nonlinear, elastoplastic damage softeningeffect and computation for polycrystal material. These simulations havedemonstrated good efficiency and accuracy.

In this exemplary example, the reduced order modeling process of wovencomposites by SCA is discussed and the results are compared with FEM.Moreover, the multiscale framework of woven composites is presented fora woven composite. Based on this framework, the part scale mechanicalresponse, whether linear or nonlinear, can be predicted efficiently onlyusing the fiber material and matrix material laws.

Methodology and Framework

SCA Method for a Woven RVE at the Mesoscale Level

A woven composite material is constructed by interweaving yarns in twodirections and then filling the weave with an epoxy matrix material. Theeffective elastic properties of an individual yarn are predicted using aUD RVE based on the constituent properties of the fiber and matrixmaterials (see FIG. 97). Then, the woven RVE is meshed by high-fidelityvoxel elements and the elastic analysis is conducted to obtain strainconcentration tensor in each element. The degrees of freedom in thewoven RVE domain are reduced by clustering these voxel elements based onthe strain concentration and orientation in each element. Using theresults of the woven RVE clustering, a material database is generatedusing the method that includes the interaction tensors D^(IJ), thestrain concentration tensor of each cluster A^(I), the volume fractionc^(I), and the material parameters of the constituents. In the onlinestage, a Newton-Raphson iteration algorithm is adopted to solve thediscrete incremental Lippmann-Schwinger integral equation set, which canimprove the accuracy and convergence, especially for nonlinear materialbehavior. The solution is the mesoscale strain and stress fields.

FEM×SCA Concurrent Multiscale Framework

Two scales, the macroscale and mesoscale, are utilized in the concurrentmultiscale framework in the study (FIG. 98). The structural scale(macroscale) is discretized by FEM, which can adapt to complexgeometries. The woven RVE scale (mesoscale) is modeled using SCA. Amultiscale simulation involving both the macroscale and mesoscale levelsis performed in which the information is exchanged concurrently.

The load is applied to the structural scale model. At each integrationpoint in the macroscale elements, the strain increment will be passedfrom the FEM model in the SCA model. This strain increment is applied tothe corresponding woven RVE, and the SCA method is used to solve thewoven RVE problem and return the stress increment to the FEM solver. Thealgorithm for the the concurrent multiscale simulation of wovencomposite structures is summarized as follows.

-   -   1. Mesh the macroscale woven composites part using FEM;    -   2. Begin solution increments;    -   3. Compute integration point field variable from nodal values;    -   4. for i=1, N_IP (Loop over integration points);        -   a. The macroscale strain increment ΔE is passed to            user-defined subroutine as input data;        -   b. Run online part of SCA to solve woven RVE subjected to ΔE            using offline woven database;        -   c. Compute the mesoscale strain increment Δe in every            cluster in woven RVE domain;        -   for j=1, N_CLU (Loop over all clusters in this corresponding            woven RVE);            -   Compute mesoscale stress increment Δσ using                corresponding material model;        -   end for (Obtain the response at all clusters);        -   d. Check convergence of the reduced-order discrete            incremental Lippmann-Schwinger integral equation, if not,            update Δe using Newton-Raphson method and go to c, if yes,            go to e;        -   e. Compute macroscale stress increment ΔΣ by averaging Δσ in            woven RVE domain and pass the macroscale stress back to the            FEM solver;    -   5. end for (Obtain the response at all integration points); and    -   6. Check convergence of the FEM part, if not, update nodal        values and go to step 3.

From the flowchart, the SCA online algorithm can be implemented by theuser-defined subroutine, which can be integrated with most commercialFEM software packages. In this way, the FEM×SCA multiscale framework canbe adapted to arbitrary structural geometry and arbitrary wovenarchitecture. Note that the cluster geometry is not required to beregular, which makes it effective for complex microstructure.

Verification of SCA for Woven RVE

Geometry Model

In the family of woven composites, plain weave composites are widelyused for ease of manufacturing. A plain weave composite is selected asan example to demonstrate and verify the SCA method at the RVE level.FIG. 99 shows the plain weave RVE microstructure used in the presentwork. The cross section of the yarns is assumed to be elliptical, andthe centerline of the yarn is modeled as a sine function. Two coordinatesystems, X₁O₁Y₁ and X₂O₂Y₂, are created to describe the cross sectionand yarn centerline features, respectively. The mathematic descriptioncan be shown in Eq. (7-1). The local coordinate frame 1-2 is used toindicate the local orientation of the yarn, which is also the localsystem the material model in yarn. The 1-direction is the tangent of thecenterline, and the 2-direction is normal to the 1-direction. This localframe varies along the length of the yarn. The woven RVE used in thisexample has 120 voxel elements in both width and length dimensions, and32 voxel elements in height dimension.

The cross sectional shape and longitudinal shape are respectivelymodeled as

$\begin{matrix}\left\{ \begin{matrix}{{\frac{x_{1}^{2}}{a^{2}} + \frac{y_{1}^{2}}{b^{2}}} = 1} \\{y_{2} = {A\;{\sin\left( {\frac{2\;\pi}{l_{0}}x_{2}} \right)}}}\end{matrix} \right. & \left( {7\text{-}1} \right)\end{matrix}$

where a is long axis, b is short axis of the elliptical cross sectionrespectively, A is amplitude of the sine function in Eq. (7-1). Thethree-dimensional geometric model of the woven RVE is defined by fiveparameters, which are also shown in FIG. 99. The values of theseparameters are assumed and only for numerical verification purposes.With proper experimental characterization, it is possible to generate arealistic woven RVE with these parameters.

Material Properties and Constitutive Model

A nonlinear epoxy plastic material model is used to model the polymermatrix. The yield function is written as

ƒ(σ,σ_(c),σ_(t))=6J ₂+2I ₁(σ_(c)−σ_(t))−2σ_(c)σ_(t)  (7-2)

where σ is Cauchy stress tensor, I₁=tr(σ) is the first invariant ofCauchy stress tensor,

$J_{2} = {\frac{1}{2}\eta\text{:}\eta}$

is the second invariant of deviatoric stress

${\eta = {\sigma - {\frac{1}{3}I_{1}}}},$

σ_(t) and σ_(c) are yield strengths in tension and compression. Anon-associative flow rule is used, with the plastic potential functionwritten as

g(σ,σ_(c),σ_(t))=6J ₂+2αI ₁(σ_(c)−σ_(t))−2σ_(c)σ_(t)  (7-3)

where

${\alpha = \frac{1 - {2v_{plas}}}{1 + v_{plas}}},$

ν_(plas) is known as plastic Poisson's ratio. Thus, the flow rule isgiven by

$\begin{matrix}{{\text{?} = {\text{?}\frac{\partial g}{\partial\sigma}}}{\text{?}\text{indicates text missing or illegible when filed}}} & \left( {7\text{-}4} \right)\end{matrix}$

where γ& represents the time derivative of the plastic multiplier. Theevolution of yield strengths in tension and compression are written as

σ_(t)=σ_(t) ₀ +H _(t)(1−e ^(−n) ^(t) ^(α) ⁰ )

σ_(c)=σ_(c) ₀ +H _(c)(1−e ^(−n) ^(c) ^(α) ¹ )  (7-5)

where σ_(t) ₀ and α_(c) ₀ are the initial yield strengths in tension andcompression, H_(t) and H_(c) are hardening parameters in case of tensionand compression respectively. These material parameters are given inTable 7-1. α₀ and α₁ are internal kinematic variables, which aredetermined by the epoxy experimental data.

TABLE 7-1 Material parameters of matrix Parameter Value E(GPa) 3.76 v0.39 v_(plas) 0.3 σ_(t) ₀ (MPa) 29 σ_(c) ₀ (MPa) 67 H_(t) (MPa) 67 H_(c)(MPa) 58 n_(t) 170 n_(c) 150

A transversely isotropic elastic material model is considered for thefibers. In addition, the elastic properties are list in Table 7-2. Thefiber volume fraction is assumed to be 60%. The subscripts 1, 2 and 3indicate the local material orientation (see FIG. 99). The UD RVE (FIG.100) used in this example has 240 voxel elements in both width andlength, 10 voxel elements in height.

TABLE 7-2 Material parameters of fiber E₁ (GPa) E₂ = E₃ (GPa) G₁₃ (GPa)v₁₂ v₂₃ Fiber 231 12.97 11.28 0.3 0.45

The effective material properties of the yarn are predicted using a UDRVE (FIG. 100) model and the elastic properties (Table 7-. 1 and Table7-2) by applying six orthogonal loads with periodic boundary conditions(PBC). As a result, the elastic material properties of yarn arepresented in Table 7-3.

TABLE 7-3 Predicted effective elastic material properties of yarn E₁(GPa) E₂ = E₃ (GPa) G₁₃ (GPa) v₁₂ v₂₃ Yarn 138.8 7.08 4.49 0.25 0 31

Clustering Process for the Woven Mesoscale RVE

The matrix material is isotropic, which requires that the clusteringonly be conducted once, based on the A_(m) tensor. After this procedure,the material points with the most similar A_(m) tensor will be groupedinto the same clusters. FIG. 101 shows the clustering results of thematrix for 256 clusters using k-means clustering.

The clustering process for the yarn material will be more complex, asillustrated in FIG. 102, where clustering process and results of yarnswith 64 clusters are shown. For each yarn, clustering is performed firstbased on local orientation. The resulting clusters are refined furtherusing strain concentration A_(m) tensor. Since one cluster correspondsone orientation dependent material law and the local frame aligns withthe points in the yarn centerline (see FIG. 99). The 1-direction istangent to the yarn centerline and represents the yarn materialorientation for each material point. The clustering process is conductedtwice. First, a single yarn is clustered based on the materialorientation using k-means. The points with the closest materialorientation will be grouped into the same cluster. Based on the resultsin the first step, the material points in the same cluster will beclustered a second time according to strain concentration tensor A_(m).In this exemplary example, two clusters are used the second time. Afterthe two-step clustering, the material points with the closestorientation and the closest strain concentration tensor will be groupedinto the same cluster. These two steps are repeated for all yarns in theRVE.

Results and Discussion

Uniaxial tension and pure shear responses are calculated using the SCAmethod. FIG. 103 and FIG. 104 include the stress-strain curves for thewoven composites under these two different load cases. The number ofclusters in the matrix ranges from 64 to 256, while the number ofclusters in the yarns is fixed at 128. When the number of clusters inthe yarns changes from 32 to 128, the number of clusters in the matrixis fixed to 256. The results from the FEM are also provided ascomparison to the baseline solution. It can be concluded that the SCAresults are in very good agreement with the FEM results under the pureshear condition in the linear and nonlinear regions. For the uniaxialtension condition, the SCA results converged with the FEM results as thenumber of clusters in the matrix increased. Since the nonlinearconstitutive model is only used for the matrix material, more clustersare needed to capture the local material nonlinear effects. Comparedwith the computational cost of FEM, SCA is capable of accuratelycapturing nonlinear behavior of woven composites with significantlyfewer degrees of freedom (SCA results differ by less than 4% comparedwith FEM). Table 7-4 and Table 7-5 present the efficiency comparisonbetween the FEM and SCA methods. FEM required about 1.45 million DOF besolved with a solution time of 6523 seconds, while SCA only requires2304 DOF and has a solution time of 30 seconds. As a result, the SCAmethod significantly improves computational efficiency in terms of timeand memory requirements. FIG. 103 shows the prediction results given byFEM and SCA (The SCA-64-128 indicates 64 clusters in matrix and 128clusters in the yarns).

TABLE 7-4 Computation efficiency comparison of FEM and SCA Elements/Computational Time/s Clusters DOF Uniaxial Tension Pure Shear FEM 4608001.45 million 6785.8 6523.1 SCA-64-128 192 1152 11.5 10.4 SCA-128-128 2561536 18.2 17.8 SCA-256-128 384 2304 28.8 19.8 Max. Speed Up 2400 1258590.1 627.2

TABLE 7-5 Computation efficiency comparison of FEM and SCA ElementsComputational Time/s Clusters DOF Uniaxial Tension Pure Shear FEM 4608001.45 million 6785.8 6523.1 SCA-256-32 288 1728 20.9 14.5 SCA-256-64 3201920 22.6 17.1 SCA-256-128 384 2304 28.8 19.8 Max. Speed Up 1600  839324.7 449.9

Property Prediction and Concurrent Multiscale Simulation

Macroscale Anisotropic Yield Surface Prediction

A nonlinear epoxy elastic-plastic material law is considered for thematrix, which results in the overall elastic-plastic behavior for wovenRVE. The yield stress of the elastic-plastic material is an importantproperty for material selection and design of composite structures. Ayield surface is developed to evaluate material yielding under variousloading conditions. The anisotropic Hill yield criterion is consideredin this example for woven composites. The homogenized material law canbe efficiently predicted using SCA based on the epoxy elastic-plasticmaterial law for the matrix and the elastic material law for the yarn.The quadratic Hill yield criterion has the following form:

F(σ_(yy)−σ_(zz))² +G(σ_(zz)−σ_(xx))² +H(σ_(xx)−σ_(yy))²+2Lσ _(yz) ²+2Mσ_(zx) ²+2Nσ _(xy) ²=1  (7-6)

where F, G, H, L, M, N are constants characteristic of the yieldsurface, which are traditionally determined by burdensome experiments.Additionally, some experiments are difficult to perform, such as theout-of-plane tension test. In this exemplary example, these parametersare predicted using the SCA method, which significantly reduces thecomputational cost and improves the efficiency. If Y_(xx), Y_(yy),Y_(zz) are the tensile yield stresses in the principal anisotropicdirection, it can be shown that:

$\begin{matrix}\begin{matrix}{{\frac{1}{Y_{x\; x}^{2}} = {G + H}},} & {{2F} = {\frac{1}{Y_{y\; y}^{2}} + \frac{1}{Y_{z\; z}^{2}} - \frac{1}{Y_{x\; x}^{2}}}} \\{{\frac{1}{Y_{y\; y}^{2}} = {H + F}},} & {{2G} = {\frac{1}{Y_{z\; z}^{2}} + \frac{1}{Y_{x\; x}^{2}} - \frac{1}{Y_{yy}^{2}}}} \\{{\frac{1}{Y_{z\; z}^{2}} = {F + G}},} & {{2H} = {\frac{1}{Y_{x\; x}^{2}} + \frac{1}{Y_{y\; y}^{2}} - \frac{1}{Y_{zz}^{2}}}}\end{matrix} & \left( {7\text{-}7} \right)\end{matrix}$

If Y_(yz), Y_(zx), Y_(xy) are the yield stresses in shear with respectto the principal axes of anisotropy, then

$\begin{matrix}\begin{matrix}{{{2L} = \frac{1}{Y_{y\; z}^{2}}},} & {{{2M} = \frac{1}{Y_{z\; x}^{2}}},} & {{2N} = \frac{1}{Y_{x\; y}^{2}}}\end{matrix} & \left( {7\text{-}8} \right)\end{matrix}$

By taking advantage of the symmetrical features of the woven RVE, onlyfour orthogonal loading conditions are applied to the RVE; the responsesare calculated using the SCA method. The tangent stiffness is computedat each point from the stress-strain response, and the yield points areidentified by evaluating the change in the tangent stiffness. As aresult, the values of yield stress in six directions are obtained. Inaddition, the Hill constants in Eq. (7-6) are calculated using Eq. (7-7)and Eq. (7-8) using the values of yield stress. The six-dimensionalyield surface descripted by Eq. (7-6) can be difficult to visualize, butby selecting three components at a time (and setting other threecomponents to be zero), this six-dimensional yield surface is plotted inthree-dimensional space, see FIG. 105 showing Hill yield surfacecalculation workflow. The 3D yield surfaces are plotted against threenormal stress components and three shear stress components. For the plotagainst normal stress components, the cross section where sigma_zz=0 isillustrated. For the plot against shear stress components, the crosssection where sigma_xy=0 is plotted.

For the nonlinear computation of woven composite structures, thisanisotropic Hill yield surface can be used as a criterion toinstantaneously identify the onset of the plastic deformation undervarious loading conditions.

The present workflow shown in FIG. 105 allows one to build yield surfacefor various microstructure and material constitutive information withminimum efforts (around one minute using a personal computer). A largewoven composite response database can be built to assist design of wovencomposite against yielding. Given a priori information on maximumservice loads, the database will provide all possible wovenmicrostructure (e.g., yarn geometry and yarn angle) and materialconstituents (e.g., matrix properties and yarn properties) that wouldprevent yielding to occur. Hence, the workflow could potentiallyaccelerate the woven composite design process by narrowing down thedesign space of various design parameters.

Multiscale Simulation Convergence Study

An RVE convergence study is first conducted to quantify the effect ofRVE size on the stress-strain response (FIG. 106). RVE-1 is a unit cellof plain weave woven composite, and RVE-2 is eight times bigger than theRVE-1. The results for these two different RVE sizes are shown in FIG.106. It is noted that the results are in close agreement with eachother, and that the RVE-1 will provide converged results with greatercomputational efficiency.

T-Shaped Hooking Structure Analysis

Woven composites are generally made of multiple layers for industrialapplication. A T-shaped hooking structure is a common geometry forconnecting different composite parts. In this example, multiscalesimulation is used to capture the macroscale and mesoscale fields indifferent layers during cyclic bending of the T-shaped hookingstructure. The structure and the loading condition are depicted in FIG.107A. Multiple layers are considered through the thickness. The redhighlighted area in FIG. 107B represents a critical zone when failurestresses are reached, as demonstrated through experiments, and a finermesh is used in this area. The total number of elements is 34,720 forthe structural level model.

For woven mesoscale RVE, 64 clusters in the matrix and 32 clusters inthe yarns are considered for the SCA calculation, while the eight-nodecontinuum brick element with a reduced integration (ABAQUS elementC3D8R) element is used for the FEM calculation. The macroscale behavioris determined by the microstructural morphologies and the mesoscaleconstitutive equation of each cluster. The SCA material database isfirst generated during the offline stage, which makes the multiscalesimulation more efficient.

This numerical study is implemented with an ABAQUS VUMAT User Subroutineand the discrete incremental Lippman-Schwinger equations are solvedusing Intel Math Kernel Library (MKL) FORTRAN codes. This numericalexample is run on Intel® Xeon® processor with 48 cores and 128 GB memory

The computational results are presented in FIG. 108. Four elements indifferent layers around the corner are selected to present the mesoscalefields. For the bending loading condition, a stress gradient existsthrough the thickness, and the stress fields are different in differentRVEs. Since the yarns have a much higher modulus, they undertake muchmore loading than matrix. The homogenized stress-strain curve at maximumstress location is plotted in FIG. 108, which shows the residual plasticstrain after loading and unloading. The stress state at peak point isplotted on the 2D yield surface, which shows that the stress has alreadyexceeded the initial yield surface. Additionally, the computationaltimes are presented in Table 7-6, which demonstrates the significantimprovements in efficiency of FEM×SCA framework.

TABLE 7-6 Computational time comparison Concurrent multiscale frameworkComputational time FE² 5.2 × 10⁵ days (Estimated) FEM × SCA 2.4 daysSpeed up 2.16 × 10⁵

The above numerical study presents the advantages using the proposedmultiscale simulation framework. The stress and strain fields can becaptured in both macroscale and mesoscale, including the nonlineareffects, which are difficult to observe using experimental technology.As a result, this framework establishes the connection between themicrostructure and macroscale response of the composites structure. Whenthe woven microstructure is modified, but the yarn and matrix materialremain unchanged, no additional experiments are needed to calibrate theconstitutive equations; only the SCA offline database needs to beupdated. In this way, it reduces the cost and improves the efficiency tofind the optimal microstructure for the specific structures. Given theefficiency of the SCA online, the larger dimensional compositesstructures can be analyzed using this framework.

In this exemplary example, a woven composite multiscale modelingframework based on Self-consistent Clustering Analysis (SCA) isestablished. A two-stage reduced order modeling for woven composite,represented as RVE, is developed: In the offline data compression stageutilized clustering technique to reduce the overall degrees of freedomin the RVE domain as material points with similar mechanical responsesare grouped into clusters. An interaction tensor linking differentclusters is computed afterwards, generating a woven RVE microstructuredatabase; In the online stage, Newton-Raphson iteration solves thereduced-order discrete incremental Lippmann-Schwinger integral equation.It exhibits rapid convergence for both linear and nonlinear materiallaws.

The woven multiscale modeling approach provides two attractivefeatures: 1) Given the woven microstructure, online stage can utilizedifferent materials laws for matrix and yarn phases to compute wovenmicrostructure responses. For example, for temperature dependentmaterial properties, the woven behavior at different operationtemperature can be computed efficiently. 2) Given the same constituentsproperties, one only needs to update the offline database to incorporatedifferent wave structures, such as plain weave, twill weave, or satinweave.

The woven multiscale modeling framework has various potentialapplication, where two important applications are illustrated in thepresent study:

Rapid yield surface generation for woven design against yielding. Theyield surface generation workflow can be used to investigate whether thewoven composite would be free of plastic deformation under possibleloading conditions. Note that it can be easily extended to failuresurface for woven design against failure.

FEM×SCA woven laminate modeling framework which captures macroscale (FEMmesh) and mesoscale mechanical behavior simultaneously during theanalysis. The mesoscale field evolution can be tracked as the loadincreases and the bridge between microstructure and macro-response isbuilt. Based on the FEM×SCA framework, the damage and failure model canbe built in the mesoscale and take more mechanisms into consideration,even conducting composites structure level failure analysis. Compared tothe traditional phenomenological constitutive relations, thesemesoscale-mechanism-based constitutive relations do not need complexmathematical formulas and numerous parameters. In addition, thisframework can be extended to larger scale structure level analysis withcomplex loading conditions. Finally, this work provides an efficientmethodology and framework to solve the woven composites multiscaleproblems.

Example 8 Self-Consistent Clustering Analysis for Multiscale Modeling atFinite Strains

Accurate and efficient modeling of microstructural interaction andevolution for prediction of the macroscopic behavior of materials isimportant for material design and manufacturing process control. Thisstudy approaches this challenge with a reduced-order method SCA. It isreformulated for general elasto-viscoplastic materials under largedeformation. The accuracy and efficiency for predicting overallmechanical response of polycrystalline materials is demonstrated with acomparison to traditional full-field solution methods such as finiteelement analysis and the fast Fourier transform. It is shown that thereduced-order method enables fast prediction of microstructure-propertyrelationships with quantified variation. The utility of the method isdemonstrated by conducting a concurrent multiscale simulation of alarge-deformation manufacturing process with sub-grain spatialresolution while maintaining reasonable computational expense. Thismethod could be used for microstructure-sensitive properties design aswell as process parameters optimization.

The process-microstructure-property chain relationship plays animportant role in the development of new materials, particularly in thepractice of computational material design or integrated computationalmaterials engineering (ICME), which relies heavily onmicrostructure-based models. These models, once calibrated, can be usedto explore a larger material design space than traditionaltrial-and-error methods. Classical micromechanics include analyticalmodels, for instance, the self-consistent method and the Mori-Tonakamethod. These models are efficient but require stringent idealizingassumptions of microstructure morphologies and/or interactions. The lastdecades have seen a rise in detailed modeling of manufacturing process,microstructure evolution and resulting mechanical properties supportedby the development of computational mechanics as well as powerfulcomputers. However, the speed of ICME deployment is still limited by thecomputational complexity of the models involved. Therefore there issubstantial interest in data-driven reduced-order models, which have thepromise of providing high accuracy without the computational expenseassociated with the detailed models used heretofore.

These reduced-order models have two important features: (1) degrees offreedom (DoFs) are significantly reduced; and (2) some data areprecalculated during offline stage so that it can be repetitively usedfor iterations during online stage. One example is the transformationfield analysis (TFA) method. It decomposes local deformation intoelastic deformation and transformation deformation (or inelasticdeformation). The elastic deformation is determined by precalculatedelastic strain concentration tensor. The transformation deformation isassumed to be uniform in each material phase. In this way, the DoFs arereduced to the order of the number of phases involved in a specificproblem. Thus this method is faster than traditional full-fieldapproaches, but fails to accurately capture intraphase heterogeneousmechanical response. Nonuniform transformation field analysis (NTFA)alleviates this problem by interpolating pre-calculated transformationfield data (or modes) under some predefined loading paths; this achieveshigher accuracy than TFA with DoFs on the order of the number of modesused. Other data-driven methods employ similar ideas to reduce DoFs, forexample proper orthogonal decomposition (POD) interpolates thedisplacement field and calculates the modes more efficiently with singlevalue decomposition. However, these methods share the same restrictions:many expensive offline calculations are needed to obtain representativemodes for highly nonlinear material behavior such as plasticity andfinite strain.

Recently, the SCA proposed by Liu et al. has been shown to maintain highaccuracy and efficiency even with these more challenging loadingconditions. To do this, SCA uses a clustering-based data compressiontechnique for order reduction and a self-consistent iterative scheme tosolve the Lippmann-Schwinger equation accurately. Some recentdevelopments of this method include theoretical analysis of convergenceof SCA, applications in toughness design of particle reinforcedcomposites, damage process of elasto-plastic strain softening materials,and fatigue life prediction for a NiTi alloy. However, most of thesestudies are limited to small strain problems and only considerinteracting microstructures with a relatively small number of features,such as homogeneous matrix, inclusions and/or voids. The finite strainSCA was used to model inclusion breakage during the drawing process. Inpractice, the constituents in the matrix phase should also be consideredif their characteristic length is comparable to inclusions and voids.For instance, metallic alloys are composed of grains with variousmorphology, size, and lattice orientation as well as possible defectssuch as precipitates or voids. Although the self-consistent andeigen-deformation based methods have been used to model interactingpolycrystals, their limitations, as outlined above, remain. We recentlycoupled a small-strain SCA formulation with a crystal plasticity model,but it neglects the rotation components of the deformation and thus isnot suited to finite strains.

This exemplary study tackles the above limitations by generalizing SCAto the finite strain case and demonstrates its accuracy and efficiencyfor predicting macroscopic mechanical response of heterogeneouselasto-viscoplastic materials, e.g., polycrystalline materials. Enabledby this approach, two case studies that would involve impractically vastcomputational time otherwise are presented for a Titanium alloy: thefirst one quantifying a microstructure-property relationship withuncertainty, and the second one predicting texture evolution of a thickplate during rolling through concurrent multiscale simulation.

The following sections discuss the data-driven mechanistic SCA methodfor finite-strain MVE problems, the accuracy and efficiency of SCA underfinite strains and compared to reference solutions using the finiteelement method (FEM) and the FFT method. The algorithm, implementation,and procedures used to generate our results are also given.

Finite-Strain Self-Consistent Clustering Analysis

Microstructural Volume Element Problem

The equilibrium mechanical response of a MVE under far-field macroscopicloading can be described by a set of equations formulated in theundeformed configuration:

$\begin{matrix}\left( {\begin{matrix}{{\frac{\partial P}{\partial X} = 0},\;{\forall{X \in \Omega}},} \\{{F = \frac{\partial u}{\partial X}},{\forall{X \in \Omega}},} \\{{\frac{1}{\Omega }{\int_{\Omega}^{\;}{F\; d\;\Omega}}} = F^{0}}\end{matrix}.} \right. & \left( \left( {8\text{-}1} \right) \right.\end{matrix}$

In these equations, P is the first Piola-Kirchhoff stress (PK1 stress),F is the deformation gradient, u is the displacement, X is a materialpoint, and Ω is the MVE domain. Pure deformation type far-field loadingF⁰ is assumed here for simplicity.

Cluster-Based Lippmann-Schwinger Equation

Under the assumption of periodic boundary conditions, the MVE problemgiven by Eq. (8-1) is equivalent to the Lippmann-Schwinger equation

F(X)+Γ⁰*(P(X)−C ⁰ :F(X))−F ⁰=0,∀X∈Ω  (8-2)

where Γ⁰ is the 4th order Green's operator associated with an arbitraryreference stiffness tensor C⁰ and * denotes convolution operationdefined by

Γ⁰*(P−C ⁰ :F)=∫_(Ω)Γ⁰(X−X′):(P(X′)−C ⁰ :F(X′))dΩ(X′).  (8-3)

To solve Eq. (8-2) numerically, a MVE domain decomposition is necessary.Unlike traditional numerical methods such as FEM and FFT, which do thisby defining a fine (relative to the minimum feature size) mesh, SCAemploys a clustering-based domain decomposition method to be introduced.Here it is assumed that the MVE is decomposed into N_(c) non-overlappingsub-domains, called clusters hereafter. For the Jth cluster Ω^(J), J=1,. . . , N_(c), the characteristic function is defined as

$\begin{matrix}{{\chi^{J}(X)} = \left( {\begin{matrix}{1,} & {{{if}\mspace{14mu} X} \in \Omega^{J}} \\{0,} & {otherwise}\end{matrix}.} \right.} & \left( {8\text{-}4} \right)\end{matrix}$

Using characteristic functions, we approximate the deformation gradientand stress as

F(X)≈Σ_(J=1) ^(N) ^(c) χ^(J)(X)F ^(J) and P(X)≈Σ_(J=1) ^(N) ^(c)χ^(J)(X)P ^(J),  (8-5)

where F^(J) is the average deformation gradient and P^(J) is the averagestress in the Jth cluster. Thus Eq. (8-6) can be approximated as

Σ_(J=1) ^(N) ^(c) χ^(J) F ^(J)+Σ_(J=1) ^(N) ^(c) (Γ⁰*χ^(J)):(P ^(J) −C ⁰:F ^(J))−F ⁰=0.  (8-6)

Multiplying both sides of Eq. (6) with χ^(I), I=1, . . . N_(c), andintegrating in Ω gives

∫_(Ω)χ^(I)(Σ_(J=1) ^(N) ^(c) χ^(J) F ^(J)+Σ_(J=1) ^(N) ^(c)(Γ⁰*χ^(J)):(P ^(J) −C ⁰ :F ^(J))−F ⁰)dΩ0.  (8-7)

The cluster-based Lippmann-Schwinger equation is obtained by simplifyingthe above equation:

F ^(I)+Σ_(J=1) ^(N) ^(c) D ^(IJ)[P ^(J) −C ⁰ :F ^(J)]−F ⁰=0, with I=1, .. . ,N _(c),  (8-8)

where D^(IJ) is the interaction tensor between the Ith cluster and Jthcluster given by

$\begin{matrix}{D^{IJ} = {\frac{1}{\Omega^{I}}{\int_{\Omega}{{\chi^{I}\left( {\Gamma^{0}*\chi^{J}} \right)}d\;{\Omega.}}}}} & \left( {8\text{-}9} \right)\end{matrix}$

Here |Ω^(I)| is the volume of the Ith cluster.

SCA solves the cluster-based Lippmann-Schwinger equation in two stages.In the offline stage, the deformation concentration tensor field (knownas the strain concentration tensor under the small strain approximation)is prepared and used to determine the clusters that define the regionsΩ^(J),J=1, . . . , N_(c) (domain decomposition), then the interactiontensors among these clusters are calculated. These data will be used inthe online stage to solve Eq. (8-8) together with local material laws.

The application of the cluster-wise approximation made in Eq. (8-5)results in the loss of deformation compatibility. This means thatalthough the solution of the continuous Lippmann-Schwinger equation, Eq.(8-2), is independent of the choice of reference stiffness tensor C⁰,the cluster-based approximation in Eq. (8-8) is not. To achieve higheraccuracy, a self-consistent iterative scheme has been proposed to updateC⁰ at each loading increment so that it approximates the effectivetangent stiffness C_(eff) of the MVE. However, this necessitatesupdating the interaction tensor during the self-consistent iterativescheme, which would increase computation time in the online stage.Fortunately, most of the interaction tensor calculation effort can bedone in the offline stage by enforcing C⁰ to be isotropi. Theformulation of C_(eff) and its isotropic approximation are given below.

Offline Stage: Micromechanical Database, Clustering and InteractionTensor Calculation

SCA reduces the degrees of freedom to be solved by taking advantage ofthe mechanical response similarity of material points in a MVE. Thissimilarity is found by clustering the field data of some mechanicalresponse. Generally, deformation concentration tensor can be used. It isdefined by

$\begin{matrix}{{{A(X)} = \frac{\partial{F(X)}}{\partial F^{0}}},{\forall{X \in {\Omega.}}}} & \left( {8\text{-}10} \right)\end{matrix}$

where F⁰ is the macroscopic deformation corresponding to the boundaryconditions of the MVE, F(X) is the local deformation at point X in theMVE domain Ω. In two dimensions, A(X) has (2×2)²=16 independentcomponents, requiring direct numerical simulations (DNS) under fourorthogonal loading conditions to uniquely define. In three dimensions,A(X) has (3×3)²=81 independent components, requiring DNS under nineorthogonal loading conditions to uniquely define. However, for specificproblems where the loading condition is known a priori, the deformationgradient (or strain) field of the same loading condition can be used forclustering. Once the clustering data is prepared, clustering methods,such as k-means clustering or self-organizing maps, can be used to finda predefined number of clusters.

The interaction tensor has to be recalculated every time C⁰ is updatedduring the self-consistent scheme. However, most of the calculationeffort can be done in the offline stage if C⁰ is isotropic. An isotropicreference stiffness tensor can be expressed as

C _(klmn) ⁰=λ⁰δ_(kl)δ_(mn)+2μ⁰δ_(km)δ_(ln),  (8-11)

where μ⁰ and λ⁰ are reference Lamé constants. The corresponding Green'soperator Γ⁰ can be decomposed into two parts:

Γ⁰ =c ₁(μ⁰,λ⁰)Γ¹ +c ₂(μ⁰,λ⁰)Γ².  (8-12)

Here, c₁ and c₂ depend on λ⁰ and μ⁰:

$\begin{matrix}\left( \begin{matrix}{{c_{1} = \frac{1}{2\mu^{0}}},{and}} \\{c_{2} = {- {\frac{\lambda^{0}}{2{\mu^{0}\left( {\lambda^{0} + {2\mu^{0}}} \right)}}.}}}\end{matrix} \right. & \left( {8\text{-}13} \right)\end{matrix}$

The terms Γ¹ and Γ² have simple forms in the Fourier frequency space:

$\begin{matrix}\left( \begin{matrix}{{{\hat{\Gamma}}_{ijkl}^{1} = \frac{\delta_{ik}\xi_{j}\xi_{l}}{{\xi }^{2}}},{and}} \\{{{\hat{\Gamma}}_{ijkl}^{2} = \frac{\xi_{i}\xi_{j}\xi_{k}\xi_{l}}{{\xi }^{4}}},}\end{matrix} \right. & \left( {8\text{-}14} \right)\end{matrix}$

where ξ is a Fourier frequency point and |ξ|=√{square root over(ξ_(i)ξ_(i))}. Thus, the interaction tensor can be expressed as

D ^(IJ) =c ₁ D ₁ ^(IJ) +c ₂ D ₂ ^(IJ).  (8-15)

where

$\begin{matrix}{{D_{w}^{IJ} = {\frac{1}{\Omega^{I}}{\int_{\Omega}{{\chi^{I}\left( {\Gamma^{w}*\chi^{J}} \right)}d\Omega}}}},{w = {1,2.}}} & \left( {8\text{-}16} \right)\end{matrix}$

Notice in Eqs. (8-14) and (8-16) that D_(w) ^(IJ), w=1,2 do not dependon the two parameters λ₀ and μ₀, thus need only be calculated once,which is done in the offline stage. If the MVE can be represented by aregular grid (i.e., voxels), the convolution in Eq. (8-16) can beobtained with relatively little computational effort using an FFTalgorithm:

Γ^(w)*χ^(J)=

⁻¹({circumflex over (Γ)}^(w)

(χ^(J))),  (8-17)

where

is the FFT operation and

⁻¹ is its inverse.

Procedure for Polycrystalline Microstructure-Property DatabaseGeneration

In certain embodiments, the implementation procedure for generating apolycrystalline microstructure-property database includes the followingsteps.

-   -   1. If not using images, set microstructure descriptors; else        -   (a) load 3D images;        -   (b) measure microstructure descriptors;    -   2. Run Dream.3D pipelines to generate M MVEs;    -   3. Initialize m=1;    -   4. Set number of clusters per grain k for MVE m;    -   5. For MVE m, run CPSCA offline calculations:        -   (a) if k=1, go to (d); else continue;        -   (b) elastic strain concentration calculation using FFT;        -   (c) domain decomposition using k-means clustering;        -   (d) interaction tensor calculation using FFT;    -   6. Set loading conditions;    -   7. Run CPSCA online subroutine;    -   8. Evaluate and add effective properties to database, and m←m+;        and    -   9. Repeat steps 4-8 until m=M.

Online Stage: Self-Consistent Scheme

For large deformation, the far field deformation gradient is appliedincrementally. The incremental far field deformation gradient ΔF⁰ of thecurrent loading step is defined by F_(current) ⁰−F_(last) ⁰. Then anincremental form of Eq. (8) is given by

ΔF ^(I)+Σ_(J=1) ^(N) ^(c) D ^(IJ):(ΔP ^(J) −C ⁰ :ΔF ^(J))−ΔF ⁰=0, forI=1, . . . ,N _(c),  (8-18)

where ΔF^(J) and ΔP^(J) are the local incremental deformation gradientand PK1 stress. Since ΔP^(J) can be determined as a function of ΔF^(J)through a local constitutive law in the Jth cluster, the unknows for Eq.(8-18) are {ΔF}={ΔF¹, ΔF², . . . , ΔF^(N) ^(c) }. The residual form ofEq. (8-18) given in Eq. (8-19) can then be solved in the online stage,using Newton's iterative method.

r ^(I) =ΔF ^(I)+Σ_(J=1) ^(N) ^(c) D ^(IJ):(ΔP ^(J) −C ⁰ :ΔF ^(J))−ΔF ⁰,for I=1, . . . ,N _(c)  (8-19)

The system Jacobian {M} is defined component-wise as:

$\begin{matrix}{{M^{IJ} = {\frac{\partial r^{I}}{{\partial\Delta}\; F^{J}} = {{\delta_{IJ}I_{4}} + {D^{IJ}\text{:}\mspace{14mu}\left( {C^{J} - C^{0}} \right)}}}},{{{for}\mspace{14mu} I,J} = 1},\ldots\;,N_{c},} & \left( {8\text{-}20} \right)\end{matrix}$

where

$C^{J} = \frac{{\partial\Delta}P^{J}}{{\partial\Delta}F^{J}}$

is the tangent stillness tensor of the material in the Jth cluster andis an output of the local constitutive law in the cluster for thecurrent loading increment. I₄ is a 4th rank identity tensor defined byI_(4,klmn)=δ_(km)δ_(ln), and δ_(IJ) is the Kronecker delta in terms ofindices I and J. The Newton's method update for the incrementaldeformation gradient can then be expressed as

{δF}=−{M} ⁻¹ {r}.  (8-21)

As mentioned, a self-consistent iterative scheme is also necessary toupdate the isotropic C⁰ at each loading increment so that itapproximates the effective tangent stiffness C_(eff) of the MVE at thatloading step. The general C_(eff) can be obtained by noting that

$\begin{matrix}{{C_{eff} = {\frac{{\partial\Delta}P^{0}}{{\partial\Delta}F^{0}} = {{\sum_{I = 1}^{N_{c}}{v^{I}\frac{{\partial\Delta}\; P^{I}}{{\partial\Delta}F^{0}}}} = {{\sum_{I = 1}^{N_{c}}{v^{I}\frac{{\partial\Delta}\; P^{I}}{{\partial\Delta}\; F^{I}}\text{:}\mspace{11mu}\frac{{\partial\Delta}\; F^{I}}{{\partial\Delta}F^{0}}}} = {\sum_{J = 1}^{N_{c}}{v^{I}C^{I}:\mspace{11mu} A^{I}}}}}}},} & \left( {8\text{-}22} \right)\end{matrix}$

where ΔP⁰ is the incremental far field PK1 stress of the current loadingstep given by ΔP⁰=Σ_(I=1) ^(N) ^(c) v^(I)ΔP^(I); v^(I) is the volumefraction of cluster I

$A^{I} = \frac{{\partial\Delta}\; F^{I}}{{\partial\Delta}F^{0}}$

is the deformation gradient concentration tensor of the Ith cluster forthe current loading increment. For each increment, this is computed bynoting that the differential form of Eq. (18) is d{ΔF⁰}={M}d{ΔF}, whichgives d{ΔF}={M}⁻¹d{ΔF⁰}, where {ΔF⁰}={ΔF⁰; . . . ; ΔF⁰} has N_(c) blocksof ΔF⁰. Denote B^(IJ) as the IJ component of the inverse of the Jacobiansystem: {B}={M}⁻¹, then A^(I)=Σ_(J=1) ^(N) ^(c) B^(IJ).

The isotropic C⁰ is obtained by minimizing ∥ΔP⁰−C⁰:ΔF⁰∥², where ∥Z∥²=Z:Zfor an arbitrary second-order tensor Z. The drawback of this method isthat the optimization problem is under-determined under pure shear orhydrostatic loading conditions. In this exemplary example, theapproximation is done by projecting the C_(eff) of the MVE to a 4th rankisotropic tensor. This is done by first expressing the isotropic C⁰ as

C ⁰=(3λ⁰+2μ⁰)J+2μ⁰ K,  (8-23)

where the forth-rank tensors J and K are defined as

J=⅓I ₂ ⊗I ₂ and K=I ₄ −J.  (8-24)

Then by noting that

J::K=0,J::J=1, and K::K=8,  (8-25)

the two Lamé constants are obtained as

$\begin{matrix}\left( \begin{matrix}{\lambda^{0} = {\frac{1}{3}\left( {{C_{eff}\text{:}\;\text{:}\mspace{11mu} J} - {\frac{1}{8}C_{eff}\text{:}\;\text{:}K}} \right)}} \\{\mu^{0} = {\frac{1}{16}C_{eff}\text{:}\;\text{:}\; K}}\end{matrix} \right. & \left( {8\text{-}26} \right)\end{matrix}$

where :: defines the double contraction operation.

Algorithm for the Online Stage of SCA

In certain embodiments, the algorithm for the online stage of SCAincludes the following steps.

-   -   1. Initial conditions and initialization:        -   (a) set n=0, {F}_(n)={I₂}, {P}_(n)=0, {ΔF}_(n)=0 and            {ΔF}_(new)={ΔF}_(n);        -   (b) call a UMAT subroutine to get C^(J), J=1, . . . , N_(c),            and set the reference stiffness C_(n) ⁰=Σ_(J=1) ^(N) ^(c)            v^(J)C^(J);    -   2. For load increment n+1, update the interaction tensor parts        {D₁} and {D₂};    -   3. Newton iterations:        -   (a) call the UMAT subroutine to get {ΔP}_(new) and {C};        -   (b) compute the residual {r};        -   (c) compute the system Jacobian {M}=∂{r}/∂{ΔF};        -   (d) solve the linear equation {M}{δF}=−{r} for {δF};        -   (e) {ΔF}_(new)←{ΔF}_(new)+{γF};        -   (f) if max_(J=1) ^(N) ^(c) {∥δF^(J)∥}<tol_(newton) is not            met, go to 3(a);    -   4. Calculate effective tangent stiffness C_(eff) and project it        to a 4th rank isotropic tensor to obtain C_(n+1) ⁰;    -   5. If ∥C_(n+1) ⁰−C_(n) ⁰∥<tol_(sc) is not met, go to step 2;    -   6. {F}_(n+1)←{F}_(n)+{ΔF}_(new), {P}_(n)←{P}_(n)+{ΔP}_(new),        n←n+1 and update the state variables for the UMAT subroutine;    -   7. Repeat steps 2-6 until the simulation completes.

Numerical Validation

By formulating the MVE problem at finite strains, we are now able tocorrectly consider general elasto-viscoplastic constitutive laws of thetype that rely on the multiplicative decomposition of the deformationgradient. One common application for such a law, commonly known ascrystal plasticity, which describes the mechanical behavior of a singlecrystal, is given. The macroscopic response of a polycrystal aggregatepredicted by SCA with such a material law is validated by comparing withtwo full-field methods. It is shown that SCA achieves comparable resultswith significantly reduced DoFs and computational expense.

Elasto-Viscoplastic Material Model

In the general elasto-viscoplastic material model, the local deformationgradient F is multiplicatively decomposed into elastic F^(e) andinelastic F^(in) contributions:

F=F ^(e) ·F ^(in).  (8-27)

F^(e) is a combination of the elastic stretch and rigid body rotation,while F^(in) is associated with unrecoverable deformation mechanisms,such as dislocation slip and/or transformation plasticity. The elasticconstitutive law is given by

S ^(e) =C ^(SE) :E ^(e)=½C _(SE):[(F ^(e))^(T) ·F ^(e) −I ₂],  (8-28)

where E^(e) is the elastic Green-Lagrange strain, S^(e) is the SecondPiola-Kirchhoff stress, C^(SE) is the 4th order elastic stiffnesstensor, and I₂ is the 2nd order identity. Taking the inelastic term assolely the plastic deformation (F^(in)=F^(p)), the inelastic deformationgradient F^(in) can be determined using a plastic constitutive law torelate the plastic velocity gradient L^(p)={dot over (F)}^(p)·(F^(p))⁻¹to plastic shear rate {dot over (γ)}^(α) in slip system a through

L ^(p)=Σ_(α=1) ^(N) ^(slip) {dot over (γ)}^(α)(s ₀ ^(α) ⊗n ₀^(α)).  (8-29)

Here, s₀ ^(α) and n₀ ^(α) are unit vectors which define the slipdirection and slip plane normal for slip system α in the undeformedconfiguration, N_(slip) is the number of active slip systems, and ⊗ isthe dyadic product. In general, the plastic shear rate {dot over(γ)}^(α) in slip system a is taken to be a function of resolved shearstress τ^(α), deformation resistance τ₀ ^(α), and back stress a^(α) inthat slip system. The resolved shear stress is given by

τ^((α))=σ:(s ^(α) ⊗n ^(α)),  (8-30)

where σ is the Cauchy stress, s^((α)) is the slip direction, and n^((α))is the slip plane normal, all of which are defined in the deformedconfiguration. They are computed from their counterparts in theundeformed configuration with

$\begin{matrix}\left( \begin{matrix}{{\sigma = {\frac{1}{J_{e}}\left\lbrack {F^{e} \cdot S^{e} \cdot \left( F^{e} \right)^{T}} \right\rbrack}},} \\{{s^{\alpha} = {F^{e} \cdot s_{0}^{\alpha}}},} \\{n^{\alpha} = {n_{0}^{\alpha} \cdot {\left( F^{e} \right)^{- 1}.}}}\end{matrix} \right. & \left( {8\text{-}31} \right)\end{matrix}$

It is possible to use any appropriate evolution law for {dot over(γ)}^(α). In this work we choose to employ a power law for {dot over(γ)}^(α) given by

$\begin{matrix}{{{\overset{.}{\gamma}}^{a} = {{\overset{.}{\gamma}}_{0}{\frac{\tau^{\alpha} - a^{\alpha}}{\tau_{0}^{a}}}^{({\overset{\sim}{m} - 1})}\left( \frac{\tau^{\alpha} - a^{\alpha}}{\tau_{0}^{\alpha}} \right)}},} & \left( {8\text{-}32} \right)\end{matrix}$

where {dot over (γ)}₀ is a reference shear rate, and {tilde over (m)} isthe exponent related to material strain rate sensitivity. The evolutionlaws for deformation resistance τ₀ ^(α) (the isotropic hardening term)and back stress a^(α) (the kinematic hardening term) are given:

$\begin{matrix}\left( \begin{matrix}{{{\overset{.}{\tau}}_{0}^{\alpha} = {{H{\sum_{\beta = 1}^{N_{slip}}{{\overset{.}{\gamma}}^{\beta}}}} - {R\;\tau_{0}^{\alpha}{\sum_{\beta = 1}^{N_{slip}}{{\overset{.}{\gamma}}^{\beta}}}}}},} \\{{\overset{.}{a}}^{\alpha} = {{h\;{\overset{.}{\gamma}}^{\alpha}} - {r\; a{{\overset{.}{\gamma}}^{\alpha}}}}}\end{matrix} \right. & \left( {8\text{-}33} \right)\end{matrix}$

where H and h are direct hardening coefficients, and R and r are dynamicrecovery coefficients. Note that in Eq. (8-33) we assume the latenthardening and self-hardening effects are identical. To account for thegrain size effect on apparent properties, a Hall-Petch-type equation isintroduced that relates the initial slip system deformation resistancein a grain, τ₀ ^(α,t=0), to the equivalent sphere diameter (ESD) D ofthat grain with

$\begin{matrix}{{\tau_{0}^{\alpha,{t = 0}} = {\tau_{0,{in}}^{\alpha,{t = 0}} + \frac{K^{\alpha}}{\sqrt{D}}}},} & \left( {8\text{-}34} \right)\end{matrix}$

where τ_(0,in) ^(α,t=0) denotes the intrinsic initial slip resistance,and K^(α) is the grain size strengthening coefficient. This equationapproximates the impeding effect of grain boundaries on dislocationslip.

Given a deformation gradient increment ΔF, its corresponding incrementof PK1 stress can be calculated following the numerical algorithm given.9 also provides the tangent stiffness

$\frac{{\partial\Delta}P}{{\partial\Delta}F}$

used to couple this material law with SCA.

Tensile Behavior of a Grain Aggregate

The material considered in this work is a fully transformed α-phaseTitanium alloy, containing 24 active hexagonal close packed (HCP) slipsystems: 3 <1120>{0001} basal, 3<1120>{1010} prismatic, 6<1120>{1011}first order pyramidal and 12<1123>{1011} second order pyramidal. Theelastic stiffness matrix is assumed to be transversely isotropic withcomponents given in Table 8-1, where C_(ij) are the stiffness componentsin Voigt notation. The material parameters for the plasticity law usedhere are given in Table 8-2; the first four entries use the calibratedvalues, and the remaining entries are assumed.

TABLE 8-1 Transversely isotropic elastic stiffness components for a HCPtitanium alloy Component Value (MPa) Component Value (MPa) C₁₁ = C₂₂1.70 × 10⁵ C₃₃ 2.04 × 10⁵ C₄₄ = C₅₅ 1.02 × 10⁵ C₆₆ 0.36 × 10⁵ C₁₂ = C₂₁0.98 × 10⁵ C₁₃ = C₃₁ 0.86 × 10⁵ C₂₃ = C₃₂ 0.86 × 10⁵ other C_(ij) 0

TABLE 8-2 Crystal plasticity parameters for a HCP titanium alloy.Pyramidal Basal Prismatic Pyramidal <a> <c + a> {dot over (γ)}₀ ^(α)(s⁻¹) * 0.0023 0.0023 0.0023 0.0023 {tilde over (m)}[*] 50 50 50 50 ^(τ)_(0, in) ^(α, t=0) (MPa) * 284.00 282.24 395.00 623.30 K^(α)(MPa√{square root over (mm)}) * 16.45 16.45 16.45 16.45 α^(α, t=0) (MPa)0.0 0.0 0.0 0.0 H (MPa) 1.0 1.0 1.0 1.0 R (MPa) 0.0 0.0 0.0 0.0 h (MPa)500.0 500.0 500.0 500.0 r (MPa) 0.0 0.0 0.0 0.0 * J Thomas, M Groeber,and S Ghosh. Image-based crystal plasticity FE framework formicrostructure dependent properties of Ti—6Al—4V alloys. MaterialsScience and Engineering: A, 553: 164-175, 2012.

The microstructure considered is an idealized MVE of eight equal-sized40 μm×40 μm×40 μm cubic grains, as shown in panel (a) of FIG. 109 ascolored according to the inverse pole FIG. 8—given in panel (b) of FIG.109, which is defined with a z-face normal as the reference direction.This allows both the FEM and FFT to use the same mesh and meshrefinement. The MVE is subjected to uniaxial tensile deformation in thex-direction with a strain rate of 10⁻⁴/sec until a maximum strain of0.02. This deformation is applied in 200 increments using three methods:SCA, FEM, and FFT; hereafter these will be termed CPSCA, CPFEM and CPFFTwhere CP is short for crystal plasticity. For CPFEM, Abaqus/Standard isused with C3D8R elements and a user material implemented as a UMAT forimplicit analysis. For CPFFT, the Newton-Krylov solver is implemented.For CPSCA, the offline clustering data is prepared with the materialmodeled as anisotropic-elastic and with the same overall loadinghistory. The elastic strain field at peak remote strain is used as theoffline data to obtain clusters using k-means clustering. Panel (c) ofFIG. 109 shows the distribution of the elastic strain component E₁₁ ofthe offline data obtained using FEM with a 10×10×10 voxel mesh. Panel(d) of FIG. 109 shows the distribution of 128 clusters (16clusters/grain) obtained using the k-means clustering method.

To show the accuracy of CPSCA, the converged macroscale stress-straincurve from full-field methods is used as the reference solution. Theconverged solution is obtained by refining the mesh to 60×60×60, whichresults in 648,000 DoFs for CPFEM and 1,944,000 for CPFFT. Panels(a)-(b) of FIG. 110 show that the elastic response is not sensitive tomesh refinement, while a substantial difference exists in the plasticdeformation regime between the results of CPFEM and CPFFT with a coursemesh, and quite similar results are achieved with a refined mesh. In thefollowing comparison, the reference solution is the CPFFT result with a60×60×60 grid, or 27,000 grid points/grain. Four different number ofclusters per grain, ranging from 1 to 64 (total DoFs ranging from 72 to4608), are used to show the convergence of CPSCA. Panels (c)-(d) of FIG.110 show that the CPSCA solution is indistinguishable (within 1%difference) even with only 1 cluster/grain in the elastic deformationregime. In the plastic deformation regime, the CPSCA solution approachesthe CPFFT solution as an increasing number of clusters is used.

CPSCA achieves high efficiency by solving only a few DoFs, withoutsacrificing much accuracy. To demonstrate this, the 0.2% plastic strainoffset (σ_(0.2), near the yield point) and 0.4% offsets stresses(σ_(0.4), away from the yield point) are compared to the referencesolution for different number of clusters. This difference is plotted inpanel (a) of FIG. 111. The difference between σ_(0.2) s as a function ofnumber of clusters oscillates, but is always below 0.4%. Conversely, thedifference between σ_(0,4) s can be as high as 1.4%, but decreasesmonotonically as the number of clusters increases. Panel (b) of FIG. 111shows that the CPU time in both offline and online stages required forCPSCA increases as a function of the number of clusters/grain used. Theslope on logarithmic axes is around two for the offline stage and aroundthree for the online stage. This is because in the offline stage thenumber of interaction tensors among the clusters that need to becomputed is N_(c) ². In the online stage, the computational complexityof matrix inversion is O(N_(c) ³). Using 64 clusters/grain (amounting to512 total clusters) takes comparable time to using CPFFT with a 40×40×40grid, but is two orders-of-magnitude faster than CPFEM using the samemesh. The increase in computational time with increasing number ofclusters is driven mostly by the online stage if a large number ofclusters are used.

Case Study 1: Uncertainty of Predicted Effective Properties

One application of microstructure modeling is to evaluate the effectiveproperties of a bulk material. For example, under monotonic tensileloading, a virtual tension test predicts the overall stress-strain curvedirectly through homogenization. From this, common scalar materialproperties such as effective elastic stiffness and yield strength can beextracted. Under cyclic loading, local stress and strain information canbe used to evaluate fatigue life. Due to restrictions on the MVE sizebecause of computational expense and microstructure randomness,uncertainty quantification of the predicted effective properties isneeded. To achieve this, multiple realizations of the MVE homogenizationare computed using the SCA method introduced above. A flowchart forgenerating a MVE-property database is shown in FIG. 112, which isimplemented with the algorithm. In this section, this procedure is usedto quantify the uncertainty of yield strength under different texturesand the grain size effect.

Synthetic Microstructure Volume Elements

Although MVEs can be obtained by extracting volumetric information from3D experimental images or grain growth simulations, as mentioned in theintroduction, such an approach could be very expensive because many MVEsare needed to generate a microstructure-property database. Morepractically, a statistical distribution of microstructure featureparameters can be measured from routine, 2D experimentalcharacterization and used to synthetically construct MVEs. To constructpolycrystalline MVEs, we use the software package DREAM.3D, whichincludes tools to generate microstructures that adhere as nearly aspossible to predetermined statistics of descriptors. This work focuseson varying two microstructure descriptors of polycrystalline materials:grain orientation distribution function and grain size distributionfunction.

The crystallographic orientation distribution function (ODF), also knownas texture, defines the probability density ƒ(Q) of crystallites fallinginto an infinitesimal neighborhood around the orientation Q, which isoften parameterized by Euler angles. In this work, we use the Bungeconvention of Euler angles (ϕ₁, Φ, ϕ₂), which define subsequentrotations about the z-axis, then the new x-axis, and then the new z-axisagain. In DREAM.3D, the orientation space is discretized into bins andan ODF with strong texture can be generated by specifying someorientations, the corresponding weights defined as multiples of randomdistribution (MRD) and number of bins it takes for the MRD value toreduce to zero decreasing quadratically from the bin of the enteredorientation.

The grain size distribution function is assumed to be log-normal withthe probability density function given by

$\begin{matrix}{{{f(D)} = {{\frac{1}{D} \cdot \frac{1}{\sigma\sqrt{2\pi}}}{\exp\left( {- \frac{\left( {{\ln D} - \mu} \right)^{2}}{2\sigma^{2}}} \right)}}},} & \text{(8-35)}\end{matrix}$

where D is the equivalent sphere diameter (ESD) of a grain, σ a scaleparameter, and μ is the shape parameter. Other descriptors such asmisorientation distribution function, aspect ratio, and number ofneighbors are also necessary to synthesize a complete microstructurerealistically

Effect of Texture on Effective Properties

To show the effect of texture on yield strength, four texture cases wereconsidered: no texture, (0,0,0) preferred, (90,45,0) preferred, and(90,90,0) preferred. Fifty cubic MVEs were synthetically generated usingDREAM.3D with grain size distribution parameters μ=19.7 μm and σ=2.7 μmfor each of the four texture cases. Each MVE has around 90 equiaxedgrains represented in a 81×81×81 voxel mesh with a resolution of 1 μm×1μm×1 μm per voxel. Example MVEs for these texture cases are given inFIG. 113. All MVEs were subjected to monotonic uniaxial tension in thex-direction with strain rate 10⁻⁴/sec until a maximum strain of 0.02.The stress-strain curves predicted using CPSCA with one cluster/grainare given in FIG. 114. The effective Young's modulus and 0.2% offsetyield strength are measured from each of these predictions and comparedin FIG. 115. MVEs with preferred (90,90,0) texture have the highestyield strength on average. MVEs with preferred (90,0,0) texture and(90,45,0) texture have approximately the same yield strength on averageand they are the lowest. MVEs without texture are in between. Thevariations in yield strength for each case are also shown.

Effect of Grain Size on Effective Properties

The effect of grain size on yield strength is studied by setting σ inEq. (8-35) such that four different average grain sizes are generatedbetween 10 μm and 40 μm. Fifty MVEs of equiaxed grains without textureare generated using DREAM.3D for each σ value while keeping all otherparameters the same. Each MVE is represented in a 81×81×81 voxel mesh.Sample MVEs with different grain size are given in FIG. 116. All MVEsare subjected to monotonic uniaxial tension in the x-direction withstrain rate 10⁻⁴/s until a maximum strain of 0.02. FIG. 117 shows thedistribution plots of predicted effective Young's modulus and 0.2%offset yield strength versus averaged ESD. The average Young's modulusis not sensitive to average ESD. A Hall-Petch type relationship betweenthe average yield strength and average ESD is also observed as expected.However, the uncertainty increases with the average ESD as the number ofgrains in the MVE decreases. This is because fewer grains makes the MVEless representative.

Case Study 2: Concurrent Multiscale Simulation of a Rolling Process

Another type of application for high-efficiency, microstructure-basedmodeling is the simulation of finite-deformation processes, in whichmicrostructure evolves extensively and gives rise to complex macroscopicbehaviors. For such applications, it is often hard to find a simple-formphenomenological constitutive model at the macroscale. Moreover, aphenomenological constitutive model has to be recalibrated for newmaterials in which microstructures are different. In this case study, itwill be shown that the speed of the CPSCA-based microstructure modelingmakes possible a concurrent multiscale model for applications wherefinite strains at the microscale are crucial. Our exemplar is the metalrolling process. Texture evolution is important at the microscale, andit depends on the macroscale loading and distribution of stressthroughout the part. To capture all this, a multiscale method isrequired.

A schematic of the rolling process of a thick plate is shown in panel(a) of FIG. 118. The initial thick plate has height (H) of 40 mm,thickness (T) of 40 mm and length (L) of 92 mm. The radius of the rolleris 170 mm with a rotation speed of 2 m/second resulting a rolling speed(roller surface speed) of 1.07 m/second. The height of the rolled plateis assumed to be 30 mm. In the initial thick plate, the microstructureis assumed to be equiaxed HCP grains with random lattice orientations(no texture) and average grain size of 26 μm. The simulation goal is topredict part deformation as well as microstructure evolution during therolling process. Direct numerical simulation of this system, by modelingall grains explicitly (more than 20 billion grains for the thick platestudied here), exceeds current modern computational capability. Analternative way is to use the two-scale concurrent multiscale simulationmethod schematically shown in panel (b) of FIG. 118. The plate(macroscale) is fully coupled with a MVE (microscale) in such a way thatthe deformation gradient of a macroscale integration point is passed toits associated MVE as far field loading, then the MVE problem is solvedand the homogenized stress is passed back to the macroscale integrationpoint. Most studies in the literature using this approach withfull-field methods e.g., FE2 for the microscale MVE problem arecomputationally limited to 2D problems. We will show in this case studythat CPSCA enables realistic 3D simulation of microstructure evolutionwith reasonable computation time.

For the macroscale problem, the implicit time integration method wasemployed so that larger time increments can be used. Coulomb frictionwas assumed between the roller and the plate, with a frictioncoefficient of 0.3 for the plate being pulled through the roll stand.The thick plate was given an initial velocity of 1.07 m/second to reduceimpact between the plate and the roller which might cause numericaldifficulty. By taking advantage of symmetry, only 1/4 plate and a singleroller were modeled. The 1/4 plate was meshed with 2994 uniformhexahedral elements and the roller was modeled with fine enough rigidelements. Reduced integration was used with stiffness-basedhour-glassing control. For the microscale problem, the MVE was chosen toinclude 90 randomly orientated equiaxed grains (panel (b) of FIG. 118)and each was represented by one cluster. The macroscale dynamic analysiswas implemented with the commercial software Abaqus/Standard and themicroscale CPSCA method was implemented as a user material subroutine,or UMAT. 8 provides the steps to calculate the stress at the end of atime step in a UMAT. The tangent stiffness tensor required byAbaqus/Standard is calculated with Eq. (8-22).

The simulation was stopped at rolling time 0.1 seconds. Panels (a)-(b)of FIG. 119 show the contours of shear stress σ₁₂ after 0.08 seconds ofrolling, predicted with the multiscale 3D simulation. For comparison,the 2D plane strain simulation result is shown in panel (c) of FIG. 119.This shows that on the symmetric plane, the 3D simulation and 2Dsimulation give similar shear stress distribution patterns: thealternating positive and negative shear stress value near the contactregion between the plate and the roller. However, the 3D simulationshows that shear stress is not uniform in the thickness direction(z-direction). The 3D simulation predicts lower deformation in therolling direction and higher extreme shear stress. The variation of σ₁₂as rolling progresses for the 3 elements indicated by the red arrows inpanel (a) of FIG. 119 is given in FIG. 120. For all 3 elements, σ₁₂alternates and reaches a stable, nonzero value in the end. The shearstress value of elements closer to the contact region tend to alternatemore times and with higher amplitude.

Another advantage of the concurrent multiscale simulation is thatmicrostructure evolution is solved for the whole manufacturing process.FIG. 121 shows snapshots of MVE deformation and (0001) pole figures(generated with MTEX) for each integration point of the three elementsduring the rolling process at the time points indicated by the verticaldashed red lines in FIG. 120. This shows that MVEs with differentposition in the macro part relative to the rolled face deform and rotatedifferently: away from the contacting surface (e.g., compare Element Ato Element C), there is less crystallographic rotation and shear, andmore compression. The rotation of each grain can be seen from the polefigures, where each dot indicates the lattice orientation of a grainplotted on a plane using the stereographic projection. During rolling,these dots concentrate around the two ends in the lateral direction ofthe circle meaning that the grain orientation rotates towards they-direction (the direction of maximum compression). This matches theexperimental observation of the texture of cold rolled, pure α-phaseTitanium. Such texture is deemed to be caused by dislocation slip whichis captured by the current model. Note that texture due to twining isdifferent. In order to predict the evolution of the Ti system moreaccurately, our model could be extended to capture twinning.

By using the Abaqus MPI parallelization with 72 cores (on three nodeseach with two Intel Haswell E5-2680v32.5 GHz 12-core processors), theconcurrent simulation takes approximately 112 h.

In sum, a finite-strain self-consistent clustering analysis is developedand applied to model interacting and evolving microstructure for generalelasto-viscoplastic materials. The method is reformulated in an initialLagrange configuration so that large deformation problems can be solved.The accuracy and efficiency obtained for the prediction of overallmechanical response of polycrystalline materials is demonstrated bycomparing with both the finite element analysis and the fast Fouriertransform-based method. It is shown that CPSCA achieves high accuracywith significantly reduced degrees of freedom.

In our case studies, grains are resolved explicitly in voxel-based MVEsreconstructed with predefined microstructure descriptors. These MVEs areused to predict with quantitative uncertainty the influence of textureon yield strength, and grain size on yield strength. Finally, aconcurrent multiscale simulation of a rolling process shows theheterogeneous microstructure evolution throughout the rolled part. Thisis made possible by the efficiency of this method. Potentialapplications include simulation-driven microstructural design andmanufacturing process control.

Example 9 Modeling and Characterization of Integrated ComputationalMaterials Engineering (ICME) Composites 1. Atomistically Informed ResinInfusion Model

In this work, predictive atomistic models of epoxy resins are developed,and the thermomechanical properties and their dependence on themolecular chemistries of the resin matrix are characterized, includingresin functionality, crosslink degree, and component ratio,demonstrating the viability of utilizing atomistic simulation to predictkey material properties and trends. In addition, we also presented ahierarchical multiscale model where MD simulation results werehomogenized to a thermo-plastic law to describe the constitutivebehavior of epoxy resins. This thermo-plastic law has been used in RVEmodeling to predict the stiffness and strength of CFRP composites.Furthermore, we characterized the properties of the nanoscale interphasebetween carbon fibers and resin matrix and integrated the interphaseinto the mechanistic continuum models for CFRP, and elucidated theexplicit effect of the interphase region on the failure behavior of thecomposites, which generated insights to guide future design strategiesfor failure-resistant composites.

The superior thermomechanical properties of epoxy resins have led to awide range of applications, most notably as matrix materials infiber-reinforced composites. The excellent thermomechanical propertiesarise from the highly crosslinked molecular structure the resins couldform. Nanoscale simulations of epoxy resins offer a promising way tocharacterize their properties and the dependence on molecular-levelfactors, such as resin type and crosslink density. Furthermore, a deepunderstanding of the dependence of thermomechanical properties on themolecular-level structures is of critical importance to guide futurecomputation-based design for epoxy resins with optimized mechanicalproperties.

Atomistic molecular dynamics (MD) simulations on epoxy resins have beensuccessfully applied to predict various material properties. Severalcomputational algorithms have been developed to generate reasonablecrosslinked structures for investigation of their physical properties.MD simulations have been carried out to predict the glass transitiontemperature (Tg) and provided valuable insights into the effects ofstrain rate, temperature, and crosslink degree on Young's modulus andyielding behavior. Despite significant progress toward understandingepoxy thermomechanical response, multiscale models that can bridgelength and time scales, especially couple atomistic and continuumscales, remains a particular challenge.

To overcome this challenge, we first developed nanoscale models ofrepresentative epoxy resins by capturing the specific crosslinkedstructures. We then characterized elastic, yield, and post-yieldbehavior from MD simulations. After that, yield surfaces were generatedfrom MD simulation results, which can be well described by a paraboloidyield criterion. Further, by adding plastic potential and hardening law,a thermo-plastic law was proposed to describe the constitutive behaviorof epoxy resins. Along the way we also illustrated the dependence ofthermomechanical properties of epoxy resins on molecular chemistry, suchas epoxy type, component ratio, and crosslink density.

In addition, the interphase region that exists between fibers and resinmatrices possesses heterogeneous chemical and physical features and hasa thickness at the sub-micron scale. Despite being much smaller than thefiber diameter, the interphase region has been shown to play a criticalrole in the performance of CFRP composites. Accurate modeling orcharacterization for the interphase region remains a significantchallenge. To overcome the hurdles encountered in nanoscopicexperiments, efforts have been reported to characterize the interphaseregion using analytical models or MD analyses. However, there have beenfew studies that integrate the nanoscale interphase region with RVEmodeling and study the effect of the interphase region on themacroscopic composite response. To address this issue, we first obtainedthe properties of the interphase region according to MD simulationresults and a generic analytical gradient model. Then, the averageproperty of the interphase region was incorporated into a modified RVEmodel, in which the three phases, fiber, matrix, and the interphase,were included. This modified RVE model was shown to improvesignificantly in predictions of the modulus and failure strength of thecomposites.

Generating the Realistic Crosslinked Structure of Epoxy Resins and YieldSurface Calculation

We chose two representative epoxy systems as our model system: (1) anepoxy resin commercially known as Epon 825, including diglycidyl etherof Bisphenol A (DGEBA) with curing agent 3,3-diaminodiphenyl sulfone(33DDS); and (2) an epoxy commercially denominated as 3501-6, mainlycomposed by tetraglycidyl methylenedianiline (TGMDA) with curing agent4,4-diaminodiphenyl sulfone (44DDS). We integrated the PolymaticAlgorithm with the LAMMPS package to simulate the crosslinking process.Basically, covalent crosslink bonds were added between eligible atomsbased on pair-wise separation distance. Also, for every severalcrosslink bonds formed, energy minimization and equilibrationsimulations were conducted with MD to alleviate the stress generated.This workflow was able to generate atomistic structures of epoxy resinswith different crosslink degrees from different initial chemistries andcomponent ratios.

To obtain the yield surface of typical epoxy resins, the stress-strainresponses of the Epon 825 model system were first calculated from the MDsimulations at different temperatures and at a strain rate of 5×10⁸ s⁻¹.We note that the high strain rate is inherent in MD simulations giventhe small time-step used. During these simulations, properthermostatting is applied to maintain the systems at specifiedtemperatures. The results for uniaxial tensile and compressive loadingcases are plotted in FIG. 122. As can be seen in the figures, the entirestress-strain response for both loads is temperature dependent,affecting the yield stresses and the elastic moduli. This behavior iswell-known in MD simulations of glassy polymers.

The subsequently obtained yield surfaces for the model system atdifferent temperatures is shown in FIG. 123. We adopted the commonconvention where yield stress corresponds to the maximum point or at theobvious “knee” in the stress-strain curve. We find that there is a goodagreement between the MD results with the paraboloidal yield surface.This yield criterion is determined uniquely by two material parameters,the compressive and tensile yield stresses:

ƒ(σ,σ_(γ) _(c) ,σ_(γ) _(T) )=6J ₂+2(σ_(c)−σ_(T))I ₁−2σ_(c)σ_(T)  (9-1)

where J₂ is the second invariant of the deviatoric stress tensor, and I₁is the first invariant of the stress tensor. σ_(T) and σ_(c) denote thetensile and compressive yielding stress, respectively.

Due to the high strain rates at which MD simulations are performed, theyield stresses obtained are higher than the values obtainedexperimentally. Nevertheless, we further find that the experimentalresults on yield stresses can be well described by the same criterion asshown in Eq. (9-1).

FIG. 123 shows yield surfaces obtained for different temperatures wherethe points are simulation data and the lines are theoretical predictionusing Eq. (9-1).

Similar to other plasticity formulations, the thermo-plastic lawdisclosed herein is then defined by the yield surface, plasticpotential, and hardening law as outlined next.

First, a plastic potential with a non-associative flow rule such that apositive volumetric plastic strain is prevented under hydrostaticpressure is defined as:

g=σ _(vm) ² +αp ²  (9-2)

where σ_(vm)=√{square root over (3J₂)} is the von Mises equivalentstress, P=1/3 I₁ is the hydrostatic pressure, and α is the materialparameter to correct the volumetric component of the plastic flow, whichequals to

$\begin{matrix}{\alpha = {\frac{9}{2}\frac{1 - {2v_{p}}}{1 + v_{p}}}} & \text{(9-3)}\end{matrix}$

with v_(p) being the plastic Poisson's ratio. This thermo-plastic law ofthe resin matrix has been integrated into the mechanistic continuummodels for CFRP with basic parameters informed by experimental results.By using this law, the characterized yield surfaces for the epoxy resinsagree very well with experimental results, with the error less than 5%,thus achieving the goal and objective of the project.

The framework for developing crosslinked epoxy resin structures as wellas yield surface characterizations is generally applicable to otherepoxy resin systems with different chemistries.

Dependence of Thermomechanical Properties on Molecular Chemistry

We have also studied the large-deformation behavior of epoxy resins andcharacterized their failure response at the atomistic level. Duringlarge deformation, there are inevitable bond breaking events happeningin the network structures of epoxy resins. To capture the realistic bondbreaking phenomena, we adopted a reactive force field, which has beenvalidated to preserve the elastic and plastic responses of the epoxyresins studied here. Stress-strain curves of 3501-6 epoxy systems withdifferent crosslink degrees and component ratios are plotted in FIG.124. Consistent “elastic-yielding-hardening-failure” behavior isobserved for all the cases. With increasing crosslink degree, both yieldand maximum stresses increase, which is associated with decreasingfailure strain or deformability. Varying the component ratio has asubtler change in the stress-strain curves, but the stoichiometric onehas the highest yield stress and maximum stress while the lowestdeformability. Thus, from atomistic level tensile simulations, we showedthat the molecular-chemistries of resins strongly influence theirmechanical properties and failure responses.

Building upon the stress-strain curves from tensile simulations and theparameters quantifying the structural changes such as chainreorientation and void formation, we have linked this atomistic levelfailure response of resins to their macroscopic fracture properties onthe basis of a continuum fracture mechanics model. This work providedphysical insights into the molecular mechanisms that govern the fracturecharacteristics of epoxy resins and demonstrated the success ofutilizing atomistic simulations toward predicting macroscopic fractureenergies.

We would like to note that the planned methodology to investigate thelarge-deformation and failure behavior of epoxy resins was to developcoarse-grained models for epoxy resins, which could increase thecomputational efficiency. We departed from the planned methodology byusing atomistic simulations with specific chemistry details captured.Although being more computationally expensive, atomistic simulationsgave us direct predictions of the thermomechanical properties whileavoiding the need to calibrate force fields for coarse-grained models.Additionally, reactive force field provides more accurate predictionsfor the stress-induced bond breaking events and failure responses of theresins at the nanoscale. More importantly, the multiscale methodology byinforming yield surface criterion and thermo-plastic constitutive lawsis more powerful to bridge length and time scales than coarse-grainedmodels.

Interphase Property Characterization

Due to the surface roughness of carbon fibers, the surface treatmentsduring fiber manufacturing process, and matrix affected regions, thereexists a submicron-thick interphase region around carbon fibers. Thethickness of the interphase region has been evaluated to be about 200 nmwith an analysis from transmission electron microscopy (TEM). Here, theinterphase region is further simplified as a cylindrical shell adjacentto the fiber, with the inner radius r_(f) being the same as the fiberradius and outer radius r_(i)=r_(f)+200 nm, as shown in FIG. 125. In thefollowing text, sub-indices ƒ, i and m denote fiber, interphase regionand matrix, respectively. Although there has not been a quantitativecharacterization of the interphase region in situ, we know some basicinformation of the property variation inside the interphase region.First, at the inner and outer boundaries, both physical and chemicalproperties of the interphase comply with the adjacent phases. Second,there exists a sharp gradient from fiber property to the matrix propertywithin the interphase region. Third, due to the incompatibility betweenthe sizing of the fiber surface and the resin matrix, we anticipate partof the regions within the interphase achieve a lower crosslink degree.The experimental observation that the failure initiates inside theinterphase region provides further evidence of this weak region. In theprevious section and FIG. 124, we have characterized the effect ofcrosslink degree on the elastic modulus and strength of typical epoxyresins. The results show that the difference of the Young's modulibetween insufficiently crosslinked epoxy (about 70% crosslink degree)and fully crosslinked epoxy (95% crosslink degree) is around 20%, andthe difference in the strength between them is up to 50%. We use E_(ms)and σ_(ms) to represent the lower bound values for the Young's modulusand strength inside the interphase region.

To characterize the average properties of the interphase region, weadopted an analytical gradient model to describe the modulus andstrength profile inside the interphase. Also, we integrated the MDsimulation results on the insufficient crosslinked resins to captureE_(ms) and σ_(ms). The gradient model proposed here include two parts.In the first part, Young's modulus and strength decrease from the fibervalues to the lowest values, i.e., E_(ms) and σ_(ms). In the secondpart, the values gradually increase from the lowest to the values of thebulk matrix. The decreasing trend in the first part is due to theattenuation of the fiber confinement effect, and the increasing trend inthe second part is because of the intrinsic epoxy resin stiffeningthrough sufficient crosslinking. We used the properties of fiber andmatrix to formulate the boundary conditions of the interphase region.The position of the lowest values (r_(is)) was assumed to be at threequarters (0.75) of the interphase away from the fiber surface. Theposition was chosen near the matrix side, since the incompatibilitybetween sizing and bulk matrix resin mainly induces the insufficientcrosslinking. A sensitivity analysis has also been conducted to verifythat the assumed position of the insufficient crosslink region has a lowinfluence on the average properties of the interphase region.

The variations of the properties of the interphase region were assumedto follow the exponential function as follows:

$\begin{matrix}{K_{i} = \left\{ \begin{matrix}{{K_{ms} + {\left( {K_{f} - K_{ms}} \right){R(r)}r_{f}}} < r < r_{is}} \\{{K_{m} + {\left( {K_{ms} - K_{m}} \right){Q(r)}r_{is}}} < r < r_{i}}\end{matrix} \right.} & \text{(9-4)}\end{matrix}$

where K can be either E (Young's modulus) or σ (strength), and thefunctions R(r) and Q(r) are constructed to match the boundaryconditions:

$\begin{matrix}{{R(r)} = \frac{1 - {\left( {r/r_{is}} \right){\exp\left( {1 - {r/r_{is}}} \right)}}}{1 - {\left( {r_{f}/r_{is}} \right){\exp\left( {1 - {r_{f}/r_{is}}} \right)}}}} & \text{(9-5)} \\{{Q(r)} = \frac{1 - {\left( {r/r_{i}} \right){\exp\left( {1 - {r/r_{i}}} \right)}}}{1 - {\left( {r_{is}/r_{i}} \right){\exp\left( {1 - {r_{is}/r_{i}}} \right)}}}} & \left( \text{9-6} \right)\end{matrix}$

The average Young's modulus and strength of the interphase can beobtained as:

K _(i)∫_(T) _(f) ^(r) ^(i) K _(i)(r)dr/(r _(i) −r _(f))  (9-7)

Substituting the parameters of both Young's modulus and strength valuesinto the above equation, we finally obtained the average Young's modulusand strength of the interphase. Compared with matrix modulus and tensilestrength, the average Young's modulus and strength of the interphaseregion are increased by around 5 and 9 times, respectively. Theinterphase region shows an obviously stiffened response compared to thebulk matrix, although a portion of the interphase region is weaker dueto insufficient crosslinking. The constitutive behavior and damage modelof the interphase were assumed to be similar to those of the bulkmatrix. By integrating this stiffened interphase region into RVE modelof the UD composites, the accuracy of RVE was much enhanced compared tothe traditional two-phase model without the interphase region. Theimportance of this interphase region is thus clearly manifested.

Our work on atomistic modeling of epoxy resins would provide guidancefor future epoxy resin computation-based design. As an importantcomponent in integrated computational materials engineering (ICME),atomistic molecular dynamics simulation would further empower thematerial-by-design process by commercially implementing the technology.

First, nanoscale simulations of highly crosslinked epoxy resins offer apromising way for the development of new continuum theories and models.Fully atomistic models are especially appealing because they are basedon fundamental input information—force fields and chemicalstructure—avoiding the need to calibrate phenomenological laws.

Second, hierarchical multiscale methods which are based on sequentialhomogenization of smaller scales to larger scales can effectivelytransfer the information from atomistic or nanoscale to the macroscopiccontinuum level. In this work, yield surface criterion has been informedfrom atomistic simulations and integrated into macroscopic models.

Third, the dependence of thermomechanical properties on molecularchemistries revealed here shows promise in accelerating thematerial-by-design process for thermosets by incorporating data frommolecular models. Potential next steps could be leveraging molecularsimulations to guide the design of the epoxy chemistry or componentratios in order to optimize the strength and toughness of the thermosetresins.

Last, we have demonstrated that by utilizing molecular simulations andanalytical models, we are able to represent the distinct interphaseregion properties between the fiber and matrix. Subsequently, we haveelucidated the explicit effect of this interphase region on the failurebehavior of composites. Building upon this, potential future work couldinvolve computational-based design strategies for failure-resistantcomposites, such as specific nano-engineered architectures andchemistries inside the interphase region.

2. Preform Molding

The preform research uses experimental and computational methods to helpunderstand the material mechanical behavior during the preformingprocess. Then based on the observation of the material behavior,preforming simulation models with high fidelity are developed. Thesemodels start from the macroscopic part-level, progress to the mesoscopiccomposite level, and finally form a multiscale simulation strategy. Themultiscale strategy enables users to have a full understanding of theprocess parameters optimization, and the lower level composite materialdesign. To validate the simulation models, preforming benchmark testswith shear angle and forming force measurement technique are developedand performed. This benchmark tests, with various combination of processparameters, provide insightful guidance to the preforming processdesign.

The traditional trial-and-error method to develop a manufacturingprocess for carbon fiber composites, which relies heavily onexperiments, requires great raw material consumption and a longdevelopment period. To solve this cost issue, the developed experimentaland computational methods for the preforming process form a wholesystem, which utilizes computation power and virtual manufacturing toolsto aid the design and optimization of the carbon fiber compositespreforming process.

The experimental research reveals the behaviors of composite prepregs inthe manufacturing process, especially the difference from theconventional metal forming process, such as temperature control, fiberreorientation, surface interaction, etc. These behaviors illustrate theimportance to adjust the manufacturing technique to the needs ofadvanced composites. The computational modeling research, on the otherhand, completes a whole software package that enables researchers fromeither academia or industry to virtually design and optimize carbonfiber prepreg preforming, which helps to lower the cost for carbon fibercomposite manufacturing development, and broaden the application of thisadvanced composite material.

To automatically manufacture CFRP parts in large quantities fortransportation equipment, thermoforming is a proper choice as it canprovide a high production rate with relatively complicated surfacegeometries, good product quality, and low facility cost. In thethermoforming process, the first step is to stack layers of thermosetcarbon fabric impregnated by uncured thermoset resin (prepreg) in anoptimized fiber orientation combination. Then, these laminates areheated to soften the resin and subsequently formed to desired 3D shapeson a press machine during the preforming step. Finally, the parts arecured to achieve designed part shapes. In thermoforming, most of thefiber re-orientation is introduced in the preforming step that replacesthe conventional high-cost and low-rate hand laying work. Sincemechanical stiffness and strength of the composites are mostly affectedby the fiber direction, the selection of the preforming parameters suchas process temperature and initial fiber orientation is important to thefinal part performance including shear and kink bands development in theweave under various loading conditions.

To optimize the preforming parameters and produce defect-free parts,numerous tests with different parameter combinations are commonlyconducted. However, the consumption of raw material and the longdevelopment period increase the cost and time of production; hamperingthe practicality of thermoforming. To address this issue, severalcomputational models have been developed to simulate the preformingprocess to predict the fiber orientation, geometry, wrinkling behavioron parts, and forming force. The first widely used computational methodto predict the woven CFRP behavior during the preforming process is thepure kinematic-based pin-joint net (PJN) assumption. However, theignorance of the mechanical properties of the fabric and the resin,results in inaccurate prediction, especially for wrinkling prediction.As an alternative, the finite element method (FEM) draws increasingattention. Simulations for the fiber orientation, draw-in amount andwrinkling behavior prediction during the preforming process have beendocumented in literatures. Jauffre et al. combined 1D beam elements and2D shell elements to simulate the tensile and shear behaviors of thematerial separately. The meshing process for this hybrid element,however, was tricky and time-consuming. Hamila et al. developed asemi-discrete triangle shell element and handled this problem based oninternal virtual work. The drawback is that this element was applicablein an in-house FEM software, limiting its usage in the industry. In theLS-DYNA® software, there are built-in woven fabric material models, suchas the MAT_234 and MAT 235. Both models, however, are based on mesoscalemechanics and require the input of mesoscale material parameters such asthe yarn moduli and yarn-yarn interaction coefficient. It was found inpractice that for these parameters, direct experimental characterizationis difficult and reverse calculation is time-consuming.

For the potential of commercialization and user-friendly operation, anon-orthogonal material model for the CFRP preforming simulation wasdeveloped by Northwestern University and was implemented into acommercial FEM code ABAQUS® as a user-defined material subroutine.Although the intention of coupling the tensile and shear behavior in thenew constitutive law was applaudable for having the most general form,it encountered inaccuracy especially when woven CFRP is subject to largeshear deformation. As an advancement, an improved non-orthogonal modelfor the woven CFRP preforming process is invented in the invention. Ithas been validated by benchmark tests and has been incorporated into theLS-DYNA® software as MAT_COMPRF (MAT_293) through the joint effort ofthis academic and industry team.

For reliable predictions, there is a need for characterizing andemploying realistic and accurate material properties in thecomputational model. During preforming, prepregs will undergo tension,shear, and bending deformation, as demonstrated in FIG. 126. FIG. 126also shows that when different prepreg layers have different initialfiber orientation, there will be large relative sliding between prepregsurfaces. As a summary, the targets of experimental characterization forprepreg are: tension, shear, bending, and surface interaction.

The most widely adopted methods for characterizing the properties withinone prepreg layer are: 1) the uniaxial tension test to determine thetensile modulus of the composites and 2) the bias-extension test tomeasure the shear modulus of the woven composites. The reliability andrepeatability of these two tests have been validated by using differentapparatuses to study a set of similar materials. The bending stiffnessof the material is also needed for proper simulation of the materialbehavior during the forming process. However, the softness of theprepreg under the preforming temperature makes it difficult to measurethe bending stiffness via the standard 3-point bending test. Analternative cantilever beam method, which utilizes self-gravity to bendspecimens, has been developed and applied to measure the bendingstiffness of prepregs. For the prepreg surface interactioncharacterization, however, a systematic study was still lacking beforethis integrated computational materials engineering (ICME) project,although this interaction surely affects the fiber orientation afterpreforming, and plays a significant role in the final productperformance. To fill this gap, we designed and built an innovative testapparatus in the invention. Based on our observation of this surfaceinteraction mechanism, a hydro-lubricant interaction model is alsoconstructed to analyze and predict the interaction between prepreglayers. This numerical model together with experiments, reveals thedetails of the fiber interaction, such as its strength and periodicpattern.

Experimental prepreg characterization can provide reliable results, butit also has some drawbacks. The major disadvantage is that it can onlyachieve limited loading states. For example, the uniaxial tension testcan only introduce pure tension deformation while the bias-extensiontest can only introduce pure shear deformation. Hence, the couplingbetween tension and shear cannot be physically characterized andsubsequently implemented into the numerical model. Although in mostcases this neglection would not affect the prediction of geometry andfiber orientation significantly due to the fact that the shear modulusof the uncured prepreg is always several orders of magnitude smallerthan its tensile modulus, it will introduce errors in the prediction ofpreforming stress and punch force and hence, reduce the analysisaccuracy of defects, such as breakage, pull-out, and separation of thefiber yarns.

Several new test devices such as the biaxial tension apparatus and thepicture frame apparatus with tension adjustment have been designed toaddress the above issue. In practice, however, even these complexdevices cannot cover all of the strain states that the prepreg willundergo during preforming due to the complexity of three-dimensionalgeometry and the resulting nonlinear loading paths. Additionally, theseexperimental characterization methods are at the macroscale and hence donot provide insightful information on how the mesoscale compositestructure and constituents affect mechanical properties of thematerials. The cost of raw materials and test time also need to beconsidered in planning experiments. To address this issue, in theinvention, we developed a new multiscale preforming simulation methodbased on the prepreg characterization by the mesoscopic representativevolume element (RVE) to account for the tension-shear coupling andapplied it to the preforming simulation of a 2×2 twill thermosetprepreg.

Challenges to building a closely packed woven RVE model includestructure generation and mesoscopic yarn properties characterization. Toconstruct RVEs with accurate woven patterns and yarn geometricalfeatures, several different approaches have been developed. One approachis to directly use CAD software to design and output the RVE structure.This approach, while being straightforward and suiTable 9—for a specificcomposite structure, is time-consuming because, for each specificcomposite, the structure needs to be drawn individually, and when theyarn surface geometries become complex, the yarn cross-section shapeneeds to be manually identified. To generalize design process andaccommodate for more composite structures, geometrical modeling softwarepackages such as TexGen and WiseTex are developed. These packages storelarge libraries for different composite patterns and can generate thecorresponding RVE structure given the geometrical features such as RVElength, yarn width, yarn height, and so on. However, the automaticallygenerated structures usually have a fixed shape of the yarncross-sections and yarn centerlines. These simplifications are suiTable9—for loose woven materials but result in yarn-to-yarn penetration inclose-packed composites. In this case, fine-tuning the geometry bymodifying the position, dimension, or utilizing non-symmetrical shapesof the local yarn cross-section is essential. These procedures, however,involve complicated geometry manipulation and are time-consuming. Forcapturing more accurate and detailed structures in RVE, researchers haverecently employed X-ray micro-tomography to directly obtain the geometryof the composites. This is a promising technique but is quite expensiveand requires careful image processing. As an alternative, to achieve afine balance between speed and accuracy in generating the RVE structure,we developed a novel 2-step geometrical modeling method in the inventionwith a one-time post-processing step to modify the local yarn geometrygenerated by TexGen.

As for the challenge to obtain unknown yarn properties at the mesoscale,a Bayesian model calibration and validation approach is developed forintegrating the calibrated mesoscale stress emulator with macroscalepart performance simulations. This is the first case in which Bayesiancalibration and hierarchical multiscale techniques are utilized forsimulation of uncured prepreg preforming.

The major technical target for the preforming modeling part of theinvention is to develop a computational simulation method that cancapture the deformation of carbon fiber composite prepregs, includingpart geometry, fiber orientation, and forming force during preforming,with high accuracy and less than 5% error. With this simulation method,the material cost and development period for design and optimization ofcarbon fiber composite prepreg preforming can be reduced significantlycompared to the conventional manufacturing process design methods, whichrely heavily on trial-and error experiments. This cost and time cut forthe development of a carbon fiber composite manufacturing process willbroaden the application of these advanced light-weight composites in thetransportation industry, making a great contribution to the control offossil fuel consumption and emission pollution.

For the success of development, experimental material characterizationtechniques are to be designed and performed systematically, first toprovide correct input to simulation models. Then models at bothmesoscale and macroscale are to be established and validated. The goalof mesoscopic modeling is to perform virtual material characterizationto replace the unsatisfactory direct experimental characterization. Thegoal of macroscopic modeling, on the other hand, is to form a platformfor part-scale preforming simulation when measured material propertiesare input. Finally, these modeling tools are to be combined with propercalibration techniques to form a high accuracy and high fidelityhierarchical multiscale modeling method for the prepreg preformingprocess.

Uncured Prepreg Characterization Experimental Methods

To obtain the well-defined constitutive model and the simulation schemesfor the numerical calculation, characterization of the uncured prepregis necessary. Several experimental methods to characterize the materialhave been developed in the invention: the uniaxial tension tests areused to characterize the tensile modulus along the fiber yarns; thebias-extension tests are used to characterize the shear modulus of theprepreg; the bending tests are employed to characterize the bendingstiffness along the yarns; and the surface interaction tests are used tocharacterize the prepreg-prepreg and prepreg-tool interaction during thepreforming process.

Uniaxial tension tests: Uniaxial tension tests are used to obtain thetensile modulus along the fiber yarns during the preforming process. Theexperiment setting is shown in FIG. 127 with the use of a tensilemachine. The mechanical properties of the uncured epoxy in the prepregis sensitive to temperature, so a temperature chamber was used here.Additionally, the digital image correlation (DIC) system was utilized tomeasure the strain distribution in samples.

In the uniaxial tension tests of the prepreg, the tensile modulus wasdetermined from the fabric tensile test, not a yarn test, so that thetensile specimen could include as many unit cells as possible. Tests atthe various temperatures that will be encountered during preforming wereconducted. To avoid slippage between the specimen and the clamps causedby the viscous epoxy, the two ends of the specimen were cured before thetests to harden the material and ensure the clamping force during thetests.

Engineering stress and strain are used to normalize the load anddisplacement data. As an example, the curves at various temperatures aredemonstrated in FIG. 128. It can be seen that as the temperatureincreases, the undulation stage of the material becomes longer, and thetensile modulus during the settle-down region slightly reduces. This isreasonable considering the softening of the epoxy at high temperature.Because of this phenomenon, the strain and the stress at the end of theundulation stage, and the stiffness after the undulation stage, wereselected to properly describe the temperature effect on the uniaxialbehavior of the prepreg.

Bias-extension tests: The bias-extension test determines the in-planeshear stiffness properties of the woven carbon fiber prepreg. Theexperiment setting is the same as that for the uniaxial tension test asshown in FIG. 127. To produce a pure shear central region, the fiberyarns were aligned ±45° off the loading direction, and the length of thespecimen should be larger than twice the width in order to release theconstraint along the yarn direction.

The temperature of the experiment was set at the various temperaturesthat will be encountered in preforming, which covers the range of thegeneral preforming process temperature measured by the IR camera duringthe single dome preforming process performed in our lab. Varioustemperatures and tensile rates are included in the tests to investigatetemperature and shear rate effects on the shear stress.

Some normalization methods are necessary for convenient utilization ofthe load-displacement data from the bias-extension tests andcompensation for the size difference of the specimen. The engineeringstress was used to normalize the load. As for the displacement, becausethe deformation is not uniform throughout the specimen, thenormalization method was derived based on the assumption that the yarnsare inextensible, and no slip occurs in the sample during thebias-extension tests. In order to validate this displacementnormalization method, two bias-extension tests with different specimensizes were performed. The parameters for the tests are listed in Table9-1. The tensile rates are selected so that the normalized tensile ratesare the same for both tests. From the results in FIG. 129, it can beseen that this displacement normalization method can properly compensatefor the specimen size difference before shear locking when the largepure shear deformation happens in the central region.

TABLE 9-1 Parameters for the displacement normalization method in thebias-extension tests. Size Temper- Test (width*thickness*length) Tensilerate ature Short 50 mm * 1.1 mm * 104.31 mm 12.00 mm/min 60° C. specimenLong 51 mm * 1.1 mm * 136.00 mm 18.87 mm/min 60° C. specimen

As an example, the normalized load-displacement curves under varioustemperatures and shear rates are plotted in FIG. 130. It can be seenthat the temperature plays an important role to the shear stress-strainrelationship, especially at the temperature near 50° C., which is thepoint where the used epoxy begins to transfer from the solid state tothe fluid state. The higher the temperature, the lower the measuredstress. The possible reason for this is that at a higher temperature,the epoxy becomes softer, reducing the resistance for the fiber warp andweft yarns to rotate relatively. As for the deformation-rate effect, ahigher deformation rate will lead to a higher final stress, which isreasonable because the deformation resistance caused by the viscosity ofthe epoxy would be larger in a higher deformation rate case. However, itshould be noted that when the deformation is smaller, the increase ofthe load happens earlier at a smaller deformation. This phenomenoncannot be explained simply by the elastic or viscous behavior of thematerial. It may require further investigating in the future.

In order to further validate the kinematical assumption that the yarnsare inextensible, and no slip occurs in the sample during thebias-extension tests, DIC technique was applied to examine the Greenstrain field in the real specimen, and the result is compared with thetheoretical one derived based on the same assumption, as shown in FIG.131. It can be seen that in the range where the DIC can work properly,the assumption holds well. When the shear deformation becomes large,however, the relative motion between the warp and weft yarns wouldscratch off the paint used for detecting the strain in the DIC systemand the result can no longer be reliable. Advanced optical methods mightbe necessary in future work in order to validate the assumption furtherand get a more precise relation between the shear deformation andstress, which can help examine the strain field of the prepreg underlarge shear deformation during the bias-extension tests.

Bending tests: The bending stiffness of the material is also needed forproper simulation of the material behavior during the forming process.However, the softness of the prepreg under the preforming temperaturemakes it difficult to measure the bending stiffness via the standard3-point bending test. As an alternative, bending tests in the inventionutilize self-gravity of the specimen. In the test, one end of therectangular prepreg sample was clamped horizontally on a support as acantilever beam, as shown in panel (a) of FIG. 132. The prepreg woulddeform under gravity due to its low rigidity in the elevatedtemperature. The deflection of the sample tip was measured, and thedeformed shape was analyzed by a digital image analysis. The entiresystem was placed in a temperature-controlled chamber, and thetemperature was recorded. The prepreg deformation during the test in a50° C. chamber temperature is given in panel (b) of FIG. 132 as ademonstration.

In this prepreg bending test, due to the strong geometric nonlinearity,the bending stiffness could not be directly calculated from the typicalbeam theory. As a solution, a simulation model was utilized to calculatethe bending stiffness reversely. This simulation model utilized thehomogeneous material properties. The compressive modulus of the materialwas modified until the same end tip displacement as the experimentalresult was reached. Then the effective compression stiffness could beobtained to properly describe the bending behavior of the prepreg.

Surface interaction tests: Before the invention, there was a lack of asystematic study for the characterization of the interaction between twodifferent composite prepreg surfaces, even though this interactionsurely affects the prepreg deformation and fiber orientation afterpreforming. To address this issue, a new experimental method wasdeveloped to characterize the interaction between uncured prepreg layersduring the preforming process. The focus of this method is tocharacterize the influence of temperature, sliding speed, and fiberorientation on the tangential interaction. The apparatus, testprocedure, and result of this new friction test method are to beillustrated specifically in the following part of this section.

The test apparatus developed in the invention is demonstrated in FIG.133. The schematic of this apparatus is shown in FIG. 134. During thetest, the top prepreg layer was clamped on the motion stage, whichresembled the relative “pull-out” slip. The bottom prepreg was clampedon a stationary heating stage, which raised the temperature between thetwo prepreg surfaces. The edges of the top prepreg layer were fixed onthe side of the motion stage instead of the bottom to avoid any possibleedge effect, and to guarantee the surface-to-surface interaction duringthe entire test. A force-torque sensor was mounted on the motion stageto record both the normal force and the tangential force caused bysliding. In this test, the compression between the two layers wasintroduced by the displacement-control motion stage. Hence, the contactforce varied during the test if the thickness of the prepreg across theentire test stage changed. Interestingly, as shown in the followinganalysis, this variation should not affect the interaction factor, whichis calculated through normalizing the in-plane tangential force by thecontact force.

An infrared (IR) camera, was also included in the system to measure thesurface temperature distribution and provided the average values of aselected area. During each test, temperature was adjusted until it couldbe maintained within +1° C. variation from the desired value bycarefully changing the power of the heating stage.

For this surface interaction test, the most important parameter is thesurface temperature because it affects the viscosity of the resin in thecomposite. During the actual preforming process, the prepreg materialwas heated to 50 about 80° C. in a heating chamber and then placed in apress that is at room temperature. Thus, the temperature selected forthe test ranges from room temperature (24° C.) to 80° C. The secondconsidered parameter is the relative motion speed, because duringpreforming, 2D sheets are deformed into 3D parts. Then the relativemotion speed between the material layers varies at different locations,resulting in a different interaction strength between the prepregs dueto the shear rate effect caused by the resin viscosity at theinterfaces. Finally, fiber orientation effect needs to be investigated.In industrial applications, prepreg layers with different fiberorientations are stacked together to optimize product performance in alldirections. Since the surface texture of the composite is anisotropic,which affects the hydrodynamic interaction between the fabric and theresin, it is also important to test the material interaction subjectedto different fiber orientation combinations.

The friction test results were analyzed when the tests with allparameter combinations were complete. The uncured prepregs are quitetacky in preforming because the resin is of high viscosity at theperforming temperature. As a result, the Coulomb friction coefficientbetween two uncured prepreg layers can be higher than 1. To avoidconfusion, the interaction factor similar to the Coulomb frictioncoefficient was defined to indicate the intensity of the yarn-resin-yarninteraction between the two prepreg layers. The results in FIG. 135 showthat at the beginning of the test, large variations in the forcecomponents and the interaction factor value were observed. The reasonsfor this include: (1) the starting point of the sliding was at the edgeof the bottom prepreg, which was not in the uniform temperature regionbecause the heating stage only provided a uniform field at the center,and (2) during the first 30 seconds of the test, the two prepreg layerswere in contact for heat conduction without relative motion. Thus, theprepregs were very tacky, resulting in a high initial interactionfactor. Based on the steady temperature field measured with the IRcamera image and the trend of the interaction factor curves, only theexperimental data within the sTable 9-stage, which is the slidingdistance ranging from 30 to 70 mm as indicated by the purple marks inFIG. 135, were utilized.

Periodical changes of the interaction factor are observed from theresulting plots, especially when the resin has very high viscosity. Takethe interaction factor at the steady-state stage shown in FIG. 136, asan example. In addition to the small oscillations caused by stick-slip,the plots from all three tests show consistent and noticeable peaks andvalleys that have similar amplitude and phase. This phenomenon will befurther investigated by the hydro-lubricant interaction model in thenext section.

The final results with respect to average stable stage temperature,relative motion speed, and fiber orientation combination are plotted inFIG. 137. Both the interaction and stick-slip strengths reach peaksvalues at 50° C., which is the critical temperature when the resin fullytransforms from the solid to the fluid state and shows the highestviscosity. Below 50° C., the resin is in its solid state. It graduallybecomes softer and tackier as the temperature increases, which leads to:(1) larger interaction factor because a stronger external force isneeded to shear the resin in different layers under the relative slidingmotion, and (2) larger amplitude and higher frequency stick-slip becauseof the more frequent molecular chain mixing and inter-diffusion, causingthe tangential interaction force to fluctuate.

The temperature effect to prepreg surface interaction is that when thetemperature is higher than 50° C., the resin fully transforms to aviscous liquid state and acts like a “lubricant” between the two prepreglayers. At this stage, further temperature increases reduce the resinviscosity and enhance its lubricity during sliding, resulting in a lowerinteraction factor, and less frequent stick-slip with a smalleramplitude. As for the relative motion speed effect, a weak influence attemperatures below 50° C. or above 60° C. was found. But the interactionfactor has a positive relation to the motion speed at about 50° C., whenthe viscosity of the resin reaches the peak value. In regard to fiberorientation effect, when the temperature is higher than 50° C., theinteraction factor becomes larger with the 0/90/0/90 fiber orientation,than that with the 0/90/−45/+45 orientation because the transverse fiberyarns in different layers are more likely to interlock with each otherwith the 0/90/0/90 orientation, making it more difficult for the layersto slide. This also explains the fact that the stick-slip strength isgenerally larger if the fibers in both top and bottom layers are alignedwith each other, especially at 50° C. when the resin viscosity reachesthe peak value. However, when the temperature is below 50° C. and theresin is in the solid state, the difference between the two orientationcombinations becomes less significant. This is mainly due to the factthat the sheets are still in the solid state, and that the resin fillsthe surface “valleys” generated by the fiber yarns to flatten theprepreg. As a result, the orientation combination does not have a largeeffect compared to those at higher temperatures.

Hydro-Lubricant Uncured Prepreg Surface Interaction Model

The woven fibers in the invention form a certain texture of surfacetopography. It has a 2×2 twill structure, as shown both by the realmaterial photo and the TexGen software model in FIG. 138. Thecharacteristic sizes of woven structure, i.e., yarn width, yarn gap, andyarn thickness, listed in Table 9-2, were measured by microscopes fromthe cross-section of the material. The average thickness of the prepregare obtained via a caliper.

TABLE 9-2 Parameters for the displacement normalization method in thebias-extension tests. Yarn width Yarn gap Yarn thickness Prepregthickness 2.430 ± 0.112 mm 0.004 ± 0.004 mm 0.503 ± 0.012 mm 0.85 ± 0.15

Textures affect the interaction of textured surfaces. A hydrodynamicmodel was developed and applied to simulate and study prepreg surfaceinteraction in the invention. In this model, the top and bottom wovenfabrics were aligned to the same direction with 0/90/0/90 fiberorientation for 2D simplification. These fabrics were treated as rigidbecause 1) they were firmly stretched in the fiber matrix, so that thevertical deformation was minimal; and 2) the normal load was low.Relative movement of the interface can be considered by the generallubrication system illustrated in FIG. 139. This system is formed withtwo solids separated by a continuous fluid film. In the simulation, thesolid geometry was determined based on the cross-section of the 2×2twill prepreg, as shown in panel (b) of FIG. 135. The single warp yarncross-section was idealized as an elliptical shape, while thecross-section of the weft yarn on top of the two warp yarns was modeledas a plane tangent to two half elliptical shapes. It was assumed that,in the simulation, the upper layer would move with respect to the lowerone. To describe the dynamic of viscous resin, one-dimensional transientReynolds equation for incompressible Newtonian fluid flow is utilized.

With this hydro-lubricant model, surface interaction at varioustemperatures is simulated at a relative motion speed set of 10 mm/s. Thecomparison of numerical and experimental results is shown in FIG. 140,where the average, maximum, and minimum values of the interaction factorare plotted. At 50° C., the numerically calculated interaction factor issignificantly larger than the experimental one because the continuityassumption is not valid. In the simulation, the resin layer behaves likea continuous fluid with high viscosity, while in the experiment, theresin may only partially melt, so there is still an interface wherefriction takes place between the top and bottom prepregs. It should benoted that, at this interface which breaks the continuity assumption inthe simulation, the friction should be lower. Moreover, in the numericalcalculation, the prepreg fiber is assumed to be rigid forsimplification. In the experiment, on the other hand, the highly viscousresin at this temperature leads to large fluid shear stress, deformingthe prepreg fiber, changing the surface profile and in return reducingthe interaction. At 60° C., the numerical results agree very well withthe experimental ones because the viscosity falls in a reasonable range,and the continuity assumption is valid since the resin fully melts. Atthe 70° C. condition, the numerical predictions are slightly smallerthan the experimental result. A larger interaction factor in theexperiment is due to the direct contact between two woven fabrics. Itwas found that at this condition, the minimum film thickness would reach0.06 mm during the calculation because the viscosity of the resinbecomes very small. The minimum film thickness is in the same order ofthe 0.012 mm half-yarn thickness variation; thus, in the real tests, twowoven fabric surfaces may contact each other at some positions,resulting in a boundary-mixed-hydrodynamic lubrication cycling.

For the interaction at 60° C., the numerical calculations with variousrelative motion speeds were then performed. The experimental andnumerical results for the average, maximum, and minimum interactionfactors are plotted in FIG. 141. The interaction model results agreewell with the experimental ones in general. However, the speed effect isslightly more significant than that found in the experiments because ofthe hydrodynamics assumption between rigid surfaces in the model, whichis sensitive to sliding speed. However, in the real experiment, otherfactors can also contribute to the speed effect. At low speed, there issufficient time for the resin to mix and inter-diffuse, so that theresin is tackier and tends to stick the two surfaces together, thusincreasing fluid resistance to motion. At high speed, the interactionforce increases because of the viscous friction, so elastic deformationof the fiber increases correspondingly, which in return flattens thesurface and reduces the interaction in the real materials.

Finally, with this hydro-lubricant model, the periodic interactionfactor variation demonstrated in FIG. 136 was investigated. The FastFourier Transformation (FFT) was applied to both experimental andnumerical results. Results at 60° C. and 10 mm/s are plotted in FIG.142, showing that all the experimental and numerical curves have the 1storder length frequency of about 0.1/mm, which means that the interactionfactor changes in the period of about 10 mm. This phenomenon isdominated by the size of the prepreg unit cell, which has a 2×2 twillelement of 9.74 mm side length. However, for higher order frequencies,numerical results agree less with experimental ones, especially in termsof amplitude. This might be explained by the fact that viscoelasticityof the real material can absorb high frequency vibration energy duringsliding.

Experiment validation demonstrates that under certain preformingconditions, i.e., 60° C. temperature and 5-15 mm/s sliding speed for thesupplied woven prepreg, interaction between two prepreg surfaces can beexplained by the hydro-lubricant mechanism and predicted via thenumerical method developed in the invention. The elastic deformation ofthe fabric and the resin mixing with inter-diffusion at variousdeformation rates and temperatures should be considered in the futurework in order to model the prepreg-prepreg interaction more accuratelyand predict the interaction behavior subjected to wider conditions.

Macroscopic Non-Orthogonal Material Model for Uncured Prepreg

In the invention a non-orthogonal material model for the CFRP preformingsimulation was developed, aiming to accurately predict the deformationof the uncured prepreg during preforming especially under large shear.This material model was developed in the form of Abaqus® explicituser-defined material subroutine (Abaqus VUMAT) and LS-DYNA® user-definematerial subroutine (LS-DYNA UMAT). Because of its ease of use and highprediction accuracy for part shape and fiber orientation, this model wasincorporated into the LS-DYNA® as MAT_293 (MAT_COMPRF). The fundamentalsof this model can be found following this section.

Woven CFRPs have highly anisotropic mechanical properties, with largetensile modulus (10 GPa level) along warp and weft yarns because of thestiff carbon fibers reinforcement, but small intra-ply shear modulus(0.1 MPa level). During preforming, the most dominant deformation modeis the intra-ply shear. To capture this fiber-orientation-dominantanisotropy, the material model needs to simulate tension along the yarnsand shear separately, even under large shear.

Stress analysis for the woven uncured prepreg with the non-orthogonalmodel developed in the invention is shown in FIGS. 143. σ_(f1), andσ_(f2) are the stress components caused by yarn stretch, and they arealong the warp and weft yarn directions, respectively. σ_(m1) and σ_(m2)are the stress components caused by the yarn rotation. These stresscomponents will be transformed into the local corotational coordinate,summed up as σ_(XX), σ_(XY), and σ_(YY), and will then be output fromthe material model to FEM software. In this model, deformation gradienttensor F is utilized to trace yarn directions and stretch ratios duringpreforming via g=F·G, where g and G are the final and initialorientations of the local fibers respectively. It can be used tocalculate α, which indicates the relative rotation between the localwarp direction and the X-direction in the local corotational coordinate,and yarn angle β, which indicates the amount of shear deformation in thematerial.

This non-orthogonal material model was implemented into both Abaqus® andLS-DYNA®. This model enables users to directly input experimental datato define the stress-strain curves, as well as the shear locking angle,which indicates whether the shear deformation reaches to the extent thatthe rotation resistance between warp and weft yarns is no longer smallcompared to the tensile modulus of the material. FIG. 144 shows thecalculation flowchart of this model in FEM software. From this flowchartit can be seen that warp and weft directions for each element arecalculated from deformation gradient tensor. If the angle between warpand weft yarns are smaller than the shear locking angle, small shearmodulus condition will hold, and total stress in the element will beupdated via the non-orthogonal model. If the angle between warp and weftyarns reaches the shear locking one, resistance for further sheardeformation will greatly increase because contacted fiber yarns stiffenthe woven structure. In this situation, the “Yarn stretch caused stress”will still be calculated via the non-orthogonal model, while the shearcomponents of the “Yarn rotation caused stress” will be updated withshear modulus of cured uni-directional (UD) carbon fiber composites.

Material characterization is essential for FEM models to accuratelypredict behavior of woven CFRPs during preforming process. It can beseen from FIG. 143 that the in-plane stresses caused by both yarnstretch and yarn rotation need to be calibrated for any specific wovenmaterial that is of interest. Their calibration can be performeddirectly by the uniaxial tension and bias-extension experiments. Theout-of-plane behaviors of the uncured prepreg, are characterized by thesingle layer bending and double layer interaction tests.

When the material model and the experimental input are prepared,double-dome benchmark tests are conducted and simulated to validate thecapability of the material model for a 3D shape forming, consideringdifferent yarn orientations and stacking sequences. These validationresults indicate that this non-orthogonal model can partly reach the 5%error target for fiber orientation prediction.

Mesoscopic RVE Model for Uncured Prepreg

Mesoscopic RVE modeling and virtual material characterization with RVErequires building of an RVE finite element model, calibration ofmesoscopic yarn properties, and generating a prepreg constitutive law asa function of strain. To build the mesh of a prepreg RVE with a finebalance between speed and accuracy, a novel 2-step geometrical modelingmethod was developed in the invention. In this method, the roughcomposite structure without yarn-to-yarn penetration is first generatedby TexGen in Step 1 with the specified woven pattern and keycharacteristic sizes, such as weaving pattern, yarn width, yarn gap, andyarn thickness. Then, the mesh and the local yarn orientationcorresponding to the structure is imported to a commercial finiteelement software in Step 2 to compress the structure in the thicknessdirection and satisfy the prepreg thickness requirements whilemaintaining the already assigned features. Finally, the deformed meshand the local material orientation are exported to build the RVE forvirtual material characterization.

This method was utilized to build the supplied 2×2 twill prepreg.Texture structure of the real 2×2 twill prepreg is shown in FIG. 138.The characteristic dimensions of the yarns and the average thicknessdata are listed in Table 9-2. Average values of yarn width, yarn gap,yarn thickness, and 2×2 twill pattern in TexGen were input first. Table9-2 indicates that the yarn gap is very small compared to yarn thicknessand width. To minimize yarn-to-yarn penetration, the shape of the yarn'scross-section was set to be lenticular. Thickness of the TexGen prepregstructure was also artificially enlarged from 0.85 mm to 1.2 mm to avoidpenetration completely. The result is illustrated in FIG. 145 and as itcan be observed, there is no longer any penetration between differentyarns.

The drawback of this thickness enlargement, however, is that many gaps,as demonstrated in panel (b) of FIG. 145, are introduced in thestructure. These artificial gaps significantly impair the predictioncapability of the RVE model; upon exerting load, the inner gaps willgreatly soften the RVE, reduce the response moduli, and elongate theundulation region. As a solution to close these gaps, the compressionmethod is introduced in Step 2. To this end, two rigid plates areemployed to compress the prepreg RVE in the thickness direction toreduce the thickness to 0.85 mm, which is the average value of the realmaterial, as illustrated in FIG. 146. The rationality of this step issupported by the fact that there is no strict constraint for yarncross-section shape and longitudinal path. At this stage, the mechanicalproperties of the prepreg yarn have not be characterized yet becausethey require calibration by the RVE, whose structure has not yet beenobtained. As a result, in the compression simulation, the elastic moduliof the yarn are selected to be the same as the existing ones for curedunidirectional composites. The Poisson's ratios are set to be zero inall directions to avoid altering the yarn width due to the yarndeformation in the thickness direction. It should be noted that theseyarn properties are only utilized to generate the RVE structure. Theyare not the same as the ones for prepreg obtained from Bayesiancalibration.

In addition to the RVE structure, the yarn material model should also becorrectly established. Because preforming is a one-step loading processwhere material recovery after the deformation is not neglected, yarnswithin RVE models are assumed purely elastic. Prepreg yarns that includequasi-unidirectional fibers and uncured resin exhibit a transverseisotropy. Direct implementation of such material behavior, however,leads to numerical errors. One kind of error happens when compressionload is applied along the width direction to a single yarn. This loadingcondition is common for prepregs in shear deformation where, asillustrated in panel (a) of FIG. 147, fibers rearrange as resin flows inreal yarns. Consequently, the yarn deforms (i.e., its dimensions change)while preserving the basic elliptical shape. In finite elementsimulation, yarns are treated as continuum with relatively flatcross-section geometry. If the transversely isotropic material model isutilized, numerical errors such as artificial bending and excessiveelement distortion will appear especially on the edges, as illustratedin panel (b) of FIG. 147. To address this issue, the transverse shearand normal behaviors in the yarn material model are decoupled to controlbending and distortion of yarns while maintaining their compressionproperty. With this approach, deformation similar to the real materialcan be achieved, as shown in panel (c) of FIG. 147.

Based on the decoupling approach, the yarn is modeled using ananisotropic elastic constitutive law with distinct Young's and shearmoduli in different directions. This constitutive law is defined in theco-rotational frame which is updated with the deformation gradienttensor to accurately trace the local fiber orientation upon large yarndeformation and rotation under the RVE deformation. In the prepregyarns, the very stiff carbon fibers are aligned in the longitudinaldirection along which the applied load is predominantly present.Meanwhile, the soft uncured resin governs the transverse deformation.Therefore, it is straightforward to decouple the yarn deformation in thelongitudinal and transverse directions.

Once the structure and the material model of the RVE are generated, theyare input into the finite element simulation given normal true strainalong yarns, shear angle, and yarn properties. After simulation, thestress of each element is extracted and averaged to obtain the stressresponse of the RVE. Mechanical properties of mesoscopic yarns includingelastic moduli, Poisson's ratios, and friction coefficient are difficultto directly characterize because of small sizes, single yarn specimenpreparation, and soft resin. As a result, the unknown materialproperties are manually adjusted at this stage and the stress predictionfrom the RVE is compared to the experimental data. One of the bestexample comparisons is illustrated in FIG. 148. The RVE result agreesvery well with the experimental one when the shear angle is less than0.6 radian, validating the 2-step approach developed. When the shearangle further increases, the discrepancy between the simulation and theexperiment becomes large, indicating the necessities for calibration.

In the invention, Bayesian calibration is applied to obtain prepreg yarnproperties for the first time. Uniaxial tension and bias-extension datais employed to: (1) estimate calibration parameters of the RVE model;(2) determine whether the RVE simulator is biased; and (3) build acheap-to-evaluate emulator to replace the expensive RVE simulation inmacroscale analyses. To this end, a modularized version of the Bayesiancalibration framework of Kennedy and O'Hagan (KOH) is adopted. The goalof Bayesian calibration is to combine three data sources (experiments,simulations, and prior knowledge from experience in the field) toestimate the unknowns. As illustrated in FIG. 149, where x representsstrain and θ represents yarn properties, this process starts byreplacing expensive mesoscopic RVE simulation with a GP emulator(metamodel) η(x, θ) in Module 1. Then, uniaxial tension experimentaldata and prior knowledge on mesoscopic yarn properties p(θ) are used tofit the GP emulator δ(x) to the bias function in Module 2. Our reasonfor introducing δ(x) is that even if true calibration parameters wereknown (which they are not) and used in simulation, the stresspredictions from the RVE model might not match with the experiments. InModule 3, joint posterior distribution of the mesoscopic yarn propertiesp(θ|d) are obtained given d, i.e., collection of the results fromexperiments and simulation. Finally, in Module 4, the updated emulatoris compared against the bias-extension experimental data for validation.Once validated, the updated emulator, as virtually characterizedconstitutive law, is utilized to predict the stress response of the RVEunder any strain state.

In practice, for supplied prepregs in the invention, uniform priordistributions (based on our experience) are chosen for θ which cover theentire range where η(x, θ) is fitted. Uniform prior distributions arepreferred (over, e.g., normal distribution) since the range of thevalues that θ can take (and not, e.g., their most likely values) areknown information. These ranges are chosen widely enough to ensure thatthe true (but unknown) calibration parameters are covered. Additionally,this choice guarantees that large variances are used to avoiddiminishing the effect of the experimental data on the joint posteriordistribution of θ.

TABLE 9-5 Prior and posterior distribution of the calibrationparameters: The priors on θ = [E₁, E₂, μ] are uniform and denoted withUni (lower bound, upper bound). Unlike the prior, the posteriordistributions of the calibration parameters are neither uniform norindependent. Prior Distribution Posterior Mode E₁ about Uni(20, 60)GPa,E₂ about Uni(5,25)MPa, μ about Uni(0.15, 3) $\quad\begin{bmatrix}{46.8\mspace{14mu}{GPa}} \\{23.5\mspace{14mu}{MPa}} \\{1.3}\end{bmatrix}$

Calibrated yarn property results are shown in FIG. 150 and demonstratethat the marginal variances are relatively large, which was expectedsince (1) there are multiple sources of uncertainty such as experimentaland simulation errors and simulator bias, and (2) limited data areemployed in the Bayesian analysis: the calibration data is only 20points from the uniaxial tension test while the RVE virtual testing goesinto complicated loading conditions.

FIG. 151 illustrates the predictions of the orthogonal stress componentsby the updated emulator under various deformation states. Normal stressσ₁₁ is plotted against normal true strain along warp and weft yarns,ε′₁₁ and ε′₂₂, for two different values of γ′₁₂ in panel (a) of FIG.151. Similarly, shear stress σ₁₂ is plotted in panel (b) of FIG. 151where its symmetry with respect to ε′₁₁ and ε′₂₂ is evident. Compared toσ₁₂, σ₁₁ is less sensitive to γ′₁₂. It can also be observed that 012monotonically increases as any of the strain components increase. Thismonotonic behavior is also observed in panel (a) of FIG. 151 but isslightly compromised when there is no shear strain (i.e., in the redsurface). This small inconsistency may be due to (1) dynamic explicitnumerical issues such as the artificially high strain rate to reduce therun-time in the RVE simulation, and (2) lack of simulation data withvery small γ′₁₂, resulting in extrapolation during the Bayesiancalibration. In panel (c) of FIG. 151, uniaxial tension experimental andpredicted results are plotted. Since this test was used for calibration,the predictions are expected to match the experiment. In panel (d) ofFIG. 151 bias extension experimental and predicted results are plotted.Since this data are not used in calibration, FIG. 151 illustrates thatthe calibration has been effective in learning the stress-strainbehavior. The posterior of the resulting GP model can now be used as theconstitutive law of integration points in the macroscopic preformingsimulations.

Multiscale Uncured Prepreg Preforming Model with Bayesian Calibration

As the constitutive law of the 2×2 twill prepreg with uncured thermosetresin, the mesoscopic stress emulator obtained from the virtual materialcharacterization is implemented into the non-orthogonal material modelto form a multiscale simulation approach. The flowchart of the developedapproach is illustrated in FIG. 152. This emulator is learned atmesoscale and acts as the non-orthogonal material constitutive law byreplacing the expensive mesoscale RVE finite element simulation at eachintegration point in macroscale preforming analysis. For the macroscopicconstitutive law in this multiscale approach, deformation input includesnormal true strain ε′₁₁ and ε′₂₂ along warp and weft yarn directions,and shear angle γ′₁₂. These inputs are all calculated using thenon-orthogonal coordinate algorithm. The predicted stress components areobtained in orthogonal material coordinate directly. Hence, theconstitutive law does not require coordinate transformation of stress.It should be noted that the prepreg stress emulator is learned over therange of ε′₁₁∈[−2, 2]%, ε′₂₂∈[−2,2]%, and γ′₁₂∈[0, 1] radian. For thedeformation states outside these ranges, the prepreg will transfer intoshear locking state, and the finite element simulation employs the shearlocking state algorithm in the non-orthogonal model.

When the multiscale preforming simulation method is established,double-dome benchmark tests are conducted and modeled to validate thecapability of the multiscale method for 3D shape forming consideringdifferent yarn orientations and stacking sequences. This validationresult reveals that this multiscale method leads to a slight improvementregarding the prediction of part geometry and fiber angle distribution,with an average of 4.0% error for fiber orientation prediction, whichachieves the proposal target. Moreover, the forming force predictionaccuracy of this multiscale method sees a significant increase of over26% compared to the experiment-based non-orthogonal model and it agreesvery well with the experimental results.

FIG. 152 shows a flowchart of the developed multiscale preformingsimulation method: The Bayesian calibration utilizes the RVE andexperiments to obtain the yarn properties and the mesoscale stressemulator. The stress emulator is then implemented into thenon-orthogonal material model for macroscopic preformation simulation.

The experimental methods to characterize mechanical properties of carbonfiber composite prepregs, the non-orthogonal material model forpreforming simulation, the mesoscopic prepreg RVE finite element modelwith Bayesian calibration for virtual material characterization, and themultiscale simulation method for preforming developed in the inventionleads to an accurate computational design and optimization approach fordevelopment of a CFRP parts preforming process. The high fidelity RVEsimulation at the material structure level also provides insightguidance for the woven pattern and constituents design in composites.This approach is able to reduce the time period and material cost forthe development of preforming compared to conventional trial-and-errormethods, which rely heavily on real experiments. As a result, thisapproach enables researchers and engineers in both academic andindustrial fields to invent and produce CFRP parts and correspondingmanufacturing processes at a faster pace, at a lower price, and in alarger volume, broadening the application of these advanced compositesand benefiting environmental emission and fossil fuel control.

The 2-step mesoscopic RVE modeling technique and multiscale simulationtool can also be commercialized and implemented into mature finiteelement software as a plug-in feature. For the RVE techniquecommercialization, the next necessary step is to integrate open TexGensoftware for the generation of fabric geometry/mesh with finite elementsoftware to form a complete package. As for commercialization of themultiscale tool, future needed steps include: (1) transferring theBayesian calibration algorithm from current MATLAB code to Python orFortran languages that can be utilized directly by finite elementsoftware; and (2) integrating RVE virtual testing model, Bayesiancalibration, and part-scale preforming simulation model into a wholepackage.

In the preforming part of the invention, we invented a set ofexperimental and simulation methods that can speed up development ofCFRP parts manufacturing at a low cost. This enables not only bigcompanies, but also smaller research teams to design their owncomposites and corresponding manufacturing processes, which potentiallyincreases applications and market needs for these advanced light-weightmaterials, reduces emission pollution from transportation industries,and motivates the carbon fiber composites production industry.

Carbon fiber composite prepregs are expensive and require specialtreatment both for storage (freezing) and for manufacturing (hightemperature to melt resin, but not cure it). Systematically arrangingphysical experiments with proper deformation and temperature checking isvery essential to avoid waste of raw material; and (2) Computationmodels, when hundreds of them need to be run for calibration, virtualmaterial characterization, and design optimization, can be timeconsuming. Validation of single simulation cases in the aspect of meshdensity, local material coordinate, material model, etc., is essentialto ensure high efficiency for large scale computational simulations. Asa summary, to save time, cut cost, and achieve neat and satisfactoryresults, either experiment or simulation research work requiressufficient preparation and planning, instead of simply performing themand relying on the trial-and-error method.

3. Mechanistic Continuum Models for CFRP

Carbon Fiber Reinforced Polymers (CFRP), including Unidirectional CFRP(UD CFRP) and twill woven CFRP (woven CFRP), have orientation dependentmaterial properties. Simply put, the tension responses of UD and WovenCFRP will be different depending on the loading condition. This is dueto the anisotropy nature of carbon fiber, which has different elasticmoduli in fiber directions and in-plane directions. Hence, to study themechanical behavior of CFRP material, it is a necessity to model CFRP'sactual microstructure for analyzing the performance of cured CFRP,including UD and woven CFRP.

The multiscale method established in the invention provides tools tomodel CFRP microstructure by the RVE method. Numerical testing of RVEspredicts effective elastic material properties of UD and woven CFRPs.Based on a reduced order modeling approach on RVEs, a concurrentmultiscale modeling method has been established to perform efficientpart-level performance prediction.

Major achievements in the UD and woven modeling are summarized as below:

-   -   User-friendly UD RVE modeling package.    -   User-friendly woven CFRP modeling package.    -   Prediction of UD elastic constants with less than 10% difference        to test data.    -   Prediction of woven elastic constants with less than 10%        difference to test data.    -   Uncertainty quantification (UQ) for Woven RVE.    -   Novel multiscale modeling concurrent with modeling for UD CFRP.    -   UD part performance predictions with less than 10% difference to        test data.    -   Woven concurrent modeling.

The properties of CFRP composites is anisotropic and microstructuredependent. Therefore, accurate capture of CFRP's mechanical properties,such as stiffness tensor, requires 1) reconstruction of themicrostructure; and 2) accurate numerical modeling. For UD CFRP, itsstiffness tensor can be simplified as the volumetric average by the ruleof mixture. It is possible to estimate the UD stiffness tensor by usingeither Voigt average or Reuss average, but the accuracy is questionablesince Voigt average gives the upper limit while Reuss average gives thelower limit. In addition, it is also proposed the modeling of UD CFRP byassuming a well-structured and periodic packing pattern of fibers, suchas hexagonal packing, and model UD CFRP by Representative Unit Cells(RUC). RUC provides easy modeling of UD CFRP since it only models theminimum repeating unit in UD CFRP and it allows different packingpatterns and fiber volume fractions. RUC can be easily modeled in finiteelement mesh and allows one to compute effective UD elastic propertieswithout dealing with algebra, compared to the analytical homogenizationapproach. Unfortunately, in real UD CFRP product, carbon fibers are of arandom distribution. Therefore, the RUC model might not provide accurateinformation about the UD's properties. For woven CFRP, RUC can be usedfor modeling mechanical properties of woven composites. However, due tothe aforementioned assumption, RUC is not the ideal way of modelingwoven CFRP when one wishes to capture certain microstructure variation.In order to include realistic microstructure, a better modelingtechnique needs to be identified and implemented so the properties of UDand woven CFRP can be predicted.

In the invention, the Representative Volume Element (RVE) approach isadopted to faithfully model UD's microstructure. The RVE model of UDcaptures the random distribution of fibers in the matrix material andcan be used for finite element analysis. Building a UD RVE allows one toconsider arbitrary fiber distribution and fiber shapes. It is expectedthat UD RVE can give a good prediction of UD's mechanical propertiescompared to test data. For woven composites, the RVE model allows one tocapture a larger region with multiple fiber tows in warp and weftdirections, and an accurate prediction of woven mechanical performancecan be made. In addition, woven RVE enables uncertainty quantificationof woven CFRP, where the effect of fiber volume fraction and fibermisalignment in fiber tow can be quantitatively analyzed. Hence, the RVEapproach provides a clear structure to the property map between the UDand woven composites and their mechanical properties. The current scopeis to use RVEs for accurate prediction of elastic stiffness tensors forboth UD and woven composites.

Moreover, the UD RVE can be applied into a recently proposed ReducedOrder Modeling (ROM) method, namely Self-consistent Clustering Analysis(SCA). SCA allows one to compress UD RVE from many voxel elements into aROM database made with several clusters. The ROM of the UD can be solvedusing the SCA method, hereafter called “UDSCA”, to computeelasto-plastic responses of UD in an efficient manner. UDSCA not onlyprovides an efficient way to compute mechanical responses (elastic andplastic) of UD, but also links UD microstructure to UD part performance.A concurrent multiscale modeling framework is established for UDmaterial for the first time and it can be used for structural propertyprediction.

The present goal of UD and Woven modeling is to develop a completemodeling workflow that allows the user to generate UD and wovenmicrostructures and extract elastic material properties, especially thestiffness tensors. It allows one to build a UD or Woven microstructureas an RVE in a finite element mesh. Microstructure information, such asUD fiber volume fraction and yarn orientation in woven composites, canbe assigned by the user. The finite element mesh can be used to performtraction free loadings in three normal directions and three sheardirections. RVE effective stress and strain results from all sixloadings will be used to compute stiffness tensor, in a 6 by 6 matrix.The present process provides direct numerical homogenization of CFRP'smaterial properties. Young's moduli and shear moduli can be computed andcompared against experimental results. The expected different between UDand Woven elastic constants from RVE models and test data is less than10%.

Moreover, an uncertainty quantification workflow is established, for thefirst time, for woven CFRP. In this workflow, variations of the wovenmicrostructure can be modeled in woven CFRP. Several microstructurevariations, such as yarn angle, fiber misalignment, and fiber volumefraction are considered. The effect of those variations on effectivewoven elastic material properties can be measured quantitatively.

For UD and Woven CFRP, an efficient reduced order modeling approach,namely self-consistent clustering analysis (SCA), is applied to reducethe computational cost of RVE computation. This allows one to computethe responses RVEs on-the-fly and enables a concurrent multiscalemodeling framework. The multiscale modeling framework establishes aconcurrent multiscale modeling framework where prediction of macroscalestructure performance is made possible. Test cases of UD CFRP structurewill be presented with experimental validation. It has broad potentialin the evaluation of CFRP structure performance through numerical modelsand can be used for future CFRP structure design.

UD RVE Modeling

In the ICME process, a bottom-up by a multiscale modeling approach isadopted for CFRP. As shown in FIG. 153, it is convenient to use athree-scale model to describe a cured CFRP part: UD in microscale, wovenin mesoscale, and part in macroscale. The RVE models for UD and wovenare built in order to evaluate their mechanical properties, such asstiffness tensors. The information obtained at lower scales can then beused in higher scales. For example, fiber tow in the woven RVE can betreated as having the same properties of UD RVE. Therefore, the UD RVEcan also be used to compute fiber tow properties that are used inevaluating woven RVE's mechanical properties. Therefore, ud modelingshould be introduced first.

An image of cured UD microstructure used in the invention is given inFIG. 154, where random fiber distribution is observed. The fiber volumemetric fraction of all UDs in the project is 51%. Hence, the modeling ofUD requires one to build a numerical model that captures random fiberdistribution with a given fiber volume fraction. For that purpose, UDRVE is used to model microstructure observed in FIG. 155. Keyassumptions for UD RVE modeling are given as below:

-   -   1) 51% overall fiber volume fraction;    -   2) Fiber has a circular shape with a diameter of 7 μm;    -   3) Fibers are perfectly straight;    -   4) RVE has a square cross-section, with a length larger than 70        μm.

The process of building a UD RVE is similar to packing multiple fibersinto a square domain. Since all fibers are assumed to be perfectlystraight, it is convenient to reduce the 3D domain into a 2D domain,where multiple circles are packed into a square. This 2D domain can beextruded in the fiber direction to form the final 3D UD RVE. Here, theUD RVE geometry is 84 μm by 84 μm by 2.8 μm, as shown in FIG. 136. InFIG. 156, fiber direction is defined as 1 direction and two transversedirections are defined as 2 and 3, following the right-hand rule. A UDRVE package is made for this particular purpose so the user can buildnew RVEs with given fiber diameter and arbitrary RVE size. The UD RVEshown in FIG. 136 is discretized by cubic voxel element with 600elements in 2 and 3 directions and 20 elements in 1 direction. When theUD RVE is used for finite element analysis, each voxel element containsonly one integration point.

The UD RVE can be used to predict elastic stiffness tensor of UD. Thematerial properties of fiber and matrix are given in Table 9-6. Sincethe primary focus is computing elastic material properties of the UD,only elastic material properties of fiber and matrix are needed.

TABLE 9-6 Fiber and matrix properties of cured woven composites FiberE₁₁ E₂₂ E₃₃ G₁₂ G₁₃ G₂₃ ν₁₂ ν₁₃ ν₂₃ 245 GPa 19.8 GPa 19.8 GPa 29.2 GPa29.2 GPa 5.92 GPa 0.28 0.28 0.32 Matrix E ν 3.79 GPa 0.39

To compute the UD stiffness matrix, it is necessary to review thedefinition of general monoclinic material. The general strain and stressrelationship of monoclinic material is given as below, where Voigtnotation is used. For UD RVE, due to its anisotropic nature, we have:S₁₂=S₂₁, S₁₃=S₃₁, S₂₃=S₃₂, S₂₂=S₃₃, S₄₄=S₅₅, S₁₄=S₄₁=0, S₂₄=S₄₂=0,S₃₄=S₄₃=0, and S₅₆=S₆₅=0. To compute individual entries of the Scompliance matrix, one needs to perform loading on the UD RVE such thatonly the stress in the loading direction is non-zero. This so-calledorthogonal loading condition allows one to compute S column by column.Therefore, loadings in 11, 22, 33, 12, 13, and 23 directions need to beperformed. Once the full compliance matrix is constructed, the stiffnesstensor is merely the inverse of the compliance matrix.

$\begin{matrix}{\begin{bmatrix}ɛ_{11} \\ɛ_{22} \\ɛ_{33} \\\gamma_{12} \\\gamma_{13} \\\gamma_{23}\end{bmatrix} = {{\begin{bmatrix}S_{11} & S_{12} & S_{13} & S_{14} & 0 & 0 \\\; & S_{22} & S_{23} & S_{24} & 0 & 0 \\\; & \; & S_{33} & S_{34} & 0 & 0 \\\; & \; & \; & S_{44} & 0 & 0 \\\; & {symmetry} & \; & \; & S_{55} & S_{56} \\\; & \; & \; & \; & \; & S_{66}\end{bmatrix}\begin{bmatrix}\sigma_{11} \\\sigma_{22} \\\sigma_{33} \\\sigma_{12} \\\sigma_{13} \\\sigma_{23}\end{bmatrix}} = {\lbrack S\rbrack_{1 - 2 - 3}\begin{bmatrix}\sigma_{11} \\\sigma_{22} \\\sigma_{33} \\\sigma_{12} \\\sigma_{13} \\\sigma_{23}\end{bmatrix}}}} & \text{(9-8)}\end{matrix}$

Once the UD RVE is generated, the mesh can be used in FE software, suchas ABAQUS, to perform loadings in six loading directions mentionedabove. Since both fiber and matrix are assumed to be elastic materials,the UD RVE responses will be strictly elastic. The UD effective stressand strain are computed using Eq. (9-9) shown below, where σ^(micro) andε^(micro) are stress and strain tensors of each voxel element in the UDRVE.

$\begin{matrix}{{\overset{¯}{\sigma} = {\frac{1}{|V|}{\int_{V}{\sigma^{micro}dV}}}},{\overset{¯}{ɛ} = {\frac{1}{|V|}{\int_{V}{ɛ^{micro}dV}}}}} & \text{(9-9)}\end{matrix}$

Once σ |ε are computed, one column of the compliance matrix can becomputed according to the specific loading direction. For example, inthe 11 loading direction, all stress components will be zero except forσ₁₁. By performing basic matrix algebra, the first column of the can becomputed entry by entry.

The quantities of interest of the UD RVE are the Young's and shearmoduli, as well as Poisson's ratios in different directions. Theconversion between UD compliance matrix and elastic moduli can be foundusing the following equivalence:

$\begin{matrix}{\begin{Bmatrix}S_{11} & S_{12} & S_{13} & \; & \; & \; \\S_{21} & S_{22} & S_{23} & \; & \; & \; \\S_{31} & S_{32} & S_{33} & \; & \; & \; \\\; & \; & \; & S_{44} & \; & \; \\\; & \; & \; & \; & S_{55} & \; \\\; & \; & \; & \; & \; & S_{66}\end{Bmatrix} = \begin{Bmatrix}\frac{1}{E_{11}} & \frac{- v_{21}}{E_{22}} & \frac{- v_{31}}{E_{33}} & \; & \; & \; \\\frac{- v_{12}}{E_{11}} & \frac{1}{E_{22}} & \frac{- v_{32}}{E_{33}} & \; & \; & \; \\\frac{- v_{13}}{E_{11}} & \frac{- v_{23}}{E_{11}} & \frac{1}{E_{33}} & \; & \; & \; \\\; & \; & \; & \frac{1}{G_{12}} & \; & \; \\\; & \; & \; & \; & \frac{1}{G_{13}} & \; \\\; & \; & \; & \; & \; & \frac{1}{G_{23}}\end{Bmatrix}} & \text{(9-10)}\end{matrix}$

The elastic material constants of UD RVE is summarized in Table 9-7,along with the experimental results. Most of the differences are allwithin 5% of the experimental data, outperforming the original proposedtarget of this work. Note that there has a relative large difference forshear moduli G12 and G13 between the experiment and prediction, but itis still within the target of 10% difference.

TABLE 9-7 Comparison of UD elastic moduli computed by UD RVE andexperimental data. E11 E22 E33 G12 G13 G23 v12 v13 v23 UD RVE 127.92 GPa8.56 GPa 8.60 GPa 4.28 GPa 4.35 GPa 2.66 GPa 0.33 0.33 0.60 Experimental133.87 GPa 8.89 GPa 8.89 GPa 4.66 GPa 4.87 GPa 2.62 GPa 0.32 0.32 N/AData Difference, % 4.44 3.80 3.25 8.17 10.68 1.45 3.16 2.75

The current package assumed fiber geometry in a circular shape, but itcan be of an arbitrary shape. Also, the present circle packing method isconsidering complete random fiber distribution, but this limits themaximum fiber volume fraction to be 60%. To achieve higher fiber volumefraction, the algorithm needs an extra function that can rearrange fiberlocations in order to exceed the 60% limit.

The generated UD RVE mesh usually has a considerable number of voxelelements, more than 1e6. Hence, UD RVE computation is most suitable forHigh Performance Computing cluster when finite element method is used.Since the UD RVE is discretized by voxel elements, this voxel mesh isessentially a 3D image. The Fast Fourier Transformation (FFT) basedhomogenization scheme is a favored algorithm to input voxel mesh andcompute the overall stress and strain responses. If the FFT basedhomogenization method is adopted, one can compute RVE elastic responsesusing a single workstation.

In short, the UD RVE model has established a convenient work-flow thatallows one to build UD RVE with desired fiber volume fraction. Allpredicted UD elastic properties either met or exceed the projectrequirement of 10% in difference.

Woven RVE

The woven RVE generation utilized TexGen, an open source software thatallows one to build a textile structure at any given pattern and fibertow (or yarn) geometry. In the invention, the woven CFRP is made oftwill woven. The minimum repeating unit of the twill woven includes fourwrap and four weft fiber yarns. The woven RVE generated is shown in FIG.156, where the woven RVE is discretized by voxel elements with aresolution of 210 by 210 by 20. Wrap yarn is in the 2 direction and weftyarn is in the 1 direction.

Once the mesh of woven RVE is generated, it can be used in FE softwareto perform numerical homogenization to obtain its elastic materialconstants. Here, the matrix material has the same material properties.Fiber yarn property is assumed to be the same as UD CFRP with 65% fibervolume fraction. Due to the aforementioned limitation, the fiber yarnproperties are computed using the analytical approach.

To analyze the elastic responses of the woven RVE, the matrix forms thestiffness tensor (due to the usage of Voigt notation) needs to becomputed using six orthogonal loading conditions, same as the UD RVE.The stiffness matrix can be computed conveniently once the compliancematrix is computed. The only difference is that for woven RVE allcomponents of the compliance matrix listed in Eq. (9-8) need to becomputed.

The computed effective elastic properties of woven RVE are listed inTable 9-8 below. All prediction met with the proposed 10% differencecompared with experimental data.

TABLE 9-8 Woven elastic moduli from woven RVE and experimental data. E11G12 G23 Woven RVE 59.96 GPa 5.68 GPa  3.6 GPa Experimental Data 65.95GPa 5.18 GPa 3.49 GPa Difference, % 9.08 9.65 3.15

The advantage of using a woven RVE numerical model is that variousmicrostructure uncertainties can be addressed in the RVE model andquantitative analyses can be done to understand the effect of thoseuncertainties. In this work, uncertainty quantification for woven CFRPis introduced for the first time. The uncertainty quantification allowsone to address uncertainties resulted from various manufacturingprocesses, such as pre-forming and curing. By giving a quantitativemeasurement of uncertainty effect, it is possible to link manufacturingprocess to the final CFRP performance, which is an important part of theICME process. Here, the woven RVE also be used to examine the effect ofthree woven microstructures as shown in FIG. 157: 1) yarn angle; 2) yarnfiber volume fraction; 3) yarn local fiber misalignment.

The effect of yarn angle is studied by constructing woven RVEs withvarious yarn angle α shown in FIG. 157. The general assumption in thepast woven CFRP research is that woven will maintain a 90° yarn angle,or the orthogonal configuration, after the performing and curingprocesses. However, it is shown that yarn angle would vary across thewoven CFRP part after the curing process. Therefore, non-orthogonalwoven RVE where yarn angle is less than 90° needs to be studied. Here,the stiffness matrices of each woven RVE realization are computed andall components are plotted in FIG. 156. Note the local materialorientation is the one shown in FIG. 156.

From FIG. 158, the yarn angle has the most significant effects on C₁₁,C₂₂, C₄₄, and C₂₄. As the yarn angle decreases, wrap yarns willgradually leaning to 1 direction and C₂₂ direction reducessignificantly. However, C₁₁ remains constant until the yarn angle isless than 60°. This means that the interaction between wrap and weftyarns is not significant when the yarn angle is larger than 60°. C₄₄tends to increase as yarn angle decreases, this means the woven in-planestiffness at a smaller yarn angle will be stronger than that at a largeryarn angle. The concave shape of C₂₄ reveals the shear-tension couplingeffect, where the in-plane shear strain will contribute to stress in 22direction when yarn angle is not 90°. All those observations reaffirmthe importance of woven yarn angle for an accurate capture of themechanical properties of woven RVE.

Besides yarn angle, yarn fiber volume fraction, denoted as V_(f), andyarn fiber misalignment effects are also investigated. The resultsobtained in Table 9-8 consider neither uncertainty, meaning the yarnmaterial is homogenous, which is usually not the case for real materialmanufactured due to manufacturing process variations. As shown in FIG.156, each yarn in the woven RVE is made with multiple voxel elements,where each voxel element contains one integration point, representingone UD RVE. Therefore, by varying fiber volume fraction at each voxelelement, inhomogeneity can be added to the yarn. By assuming fibervolume fraction on each yarn is following a Gaussian distribution, it ispossible to assign different UD properties in different voxel elementsto simulation different V_(f) in the yarn. The effect of V_(f) issummarized in Table 9-9, where V_(f) is following a Gaussiandistribution with mean V _(f)=65% and variance σ_(v) _(f) ²=0.09%. Itcan be concluded that the effect of σ_(v) _(f) ²=0.09% does not affectwoven property significantly, primarily depending on its mean value.

TABLE 9-9 Effect of yarn fiber volume fraction on homogenized materialproperties. E₁₁ (GPa) E₂₂ (GPa) E₃₃(GPa) G₁₂(GPa) G₁₃(GPa) G₂₃(GPa)V_(f) = 65% 59.96 59.96 12.57 5.68 3.60 3.60 V _(f) = 65%, 59.96 59.9412.60 5.66 3.58 3.58 σ_(V) _(f) ² = 0.09% Percentage 0.0011 0.02180.2128 0.4066 0.4436 0.4425 Difference (%)

Moreover, each voxel element contains a local material orientation thataligns with the yarn center line for homogeneous material. Fibermisalignment is considered as the deviation from perfect alignmentdirection. Shown in FIG. 157, vector {right arrow over (g)}₁ representsthe direction of perfect fiber direction, which is essentially thetangent line of yarn center line. Plane {right arrow over (g)}₂{rightarrow over (G)}₃ is the yarn cross-section and {right arrow over (g)}₁is orthogonal to the plane.

Angle θ (0°≤θ≤90°) and Φ (−180°≤Φ≤180°) are used to establish misalignedfiber direction {right arrow over (f)}₁, {right arrow over (f)}₁, {rightarrow over (f)}₂, and {right arrow over (f)}₃ represent transverseisotropic material frame accounting for fiber misalignment. Equationsfor calculating {right arrow over (f)}₁, {right arrow over (f)}₂, and{right arrow over (f)}₃ given as below:

$\begin{matrix}{{{\overset{\rightarrow}{f}}_{1} = {{\frac{{\overset{\rightarrow}{g}}_{2}}{{\overset{\rightarrow}{g}}_{2}}\sin\theta\cos\Phi} + {\frac{{\overset{\rightarrow}{g}}_{3}}{{\overset{\rightarrow}{g}}_{3}}{\sin{\theta sin\Phi}}} + {\frac{{\overset{¯}{g}}_{1}}{{\overset{\rightarrow}{g}}_{1}}{\cos\theta}}}}{{\overset{\rightarrow}{f}}_{2} = {{\frac{{\overset{\rightarrow}{g}}_{2}}{{\overset{\rightarrow}{g}}_{2}}\cos\theta\cos\Phi} + {\frac{{\overset{\rightarrow}{g}}_{3}}{{\overset{\rightarrow}{g}}_{3}}\cos\theta\sin\Phi} - {\frac{{\overset{\rightarrow}{g}}_{1}}{{\overset{\rightarrow}{g}}_{1}}\sin\theta}}}{{\overset{\rightarrow}{f}}_{3} = {{\overset{\rightarrow}{f}}_{1} \times {\overset{\rightarrow}{f}}_{2}}}} & \text{(9-11)}\end{matrix}$

For fiber misalignment, θ and φ follow gaussian distribution by lettingmean θ=10°, variance σ_(θ) ²=2, mean φ=0°, and variance σ_(φ) ²=2500°²(to make sure for all element within the yarn, its φ will fall between−180° and 180° following three sigma rule).

In summary, woven RVE modeling provides a straightforward numericalanalysis tool for studying woven CFRP mechanical properties, where allpredictions are within 10% difference compared to test data. Moreover,the test cases of woven uncertainty parameters illustrate themicrostructural effect in woven CFRP. Woven UQ provides a convenientnumerical solution to evaluate possible uncertainties caused bydifferent manufacturing processes.

Reduced Order Modeling of UD CFRP

Aforementioned ud package is able to generate a UD RVE in voxel mesh andallow the user to analyze the mechanical responses of UD CFRP. However,the high computational cost associated with the fine voxel mesh requirescertain reduced order model (ROM) techniques to achieve 1) faster RVEresponses computation; 2) linking UD RVE to large-scale part-level modelfor part performance prediction with experimental validation (within 10%difference).

TABLE 9-10 Effect of Fiber Misalignment on Homogenized materialproperties. E₁₁ (GPa) E₂₂(GPa) E₃₃ (GPa) G₁₂ (GPa) G₁₃ (GPa) G₂₃ (GPa)No- 59.96 59.96 12.57 5.68 3.60 3.60 Misalignment θ = 10°, 48.01 47.1412.42 6.15 3.67 3.68 σ_(θ) ² = 2 Φ = 0°, σ_(Φ) ² = 2500 Percentage 24.8927.19 1.17 7.69 1.88 2.15 Difference (%)

A recently proposed reduced order modeling method, namelyself-consistent clustering analysis (SCA), is a promising method forbuilding ROM for arbitrary voxel mesh, including UD RVE. It is based onthe FFT homogenization scheme. In FFT homogenization scheme, straintensor at each voxel is treated as a combination of overall imposedstrain ε^(Macro) and a polarization strain {tilde over (ε)}, shown inEq. (9-14) below

ε(X)={tilde over (ε)}+ε^(Macro)  (9-14)

Above equation, also known as Lipmman-Schwinger equation, allows one tosolve local strain responses ε(X) when ε^(Macro) is fixed. This is thebasic of fft homogenization method, which is time consuming since theevaluation happens for all voxel elements. Eq. (9-3) can also be writtenas Eq. (9-15) as below

ε^(Macro)−ε(X)−∫_(Ω)Γ⁰(X,X′)_(:[σ(X′)−C) ⁰:ε(X′)]dX′=0,X∈Ω  (9-15)

Liu et. al. proposed a reduced order modeling approach byre-discretizing the voxel mesh into several clusters. Assuming theoriginal voxel mesh contains N elements, the mesh can be compressed intoK clusters, where N>>K. Eq. (15) is reformulated as Eq. (9-16) as shownbelow.

$\begin{matrix}{{ɛ^{Macro} - ɛ^{I} - {\sum\limits_{J = 1}^{K}{D^{IJ}:\left\lbrack {\sigma^{J} - {C^{0}:ɛ^{J}}} \right\rbrack}}} = 0} & \text{(9-16)}\end{matrix}$

Eq. (9-16) can be easily solved using Newton's method. Since N>>K, Eq.(9-5) is a much smaller linear system to solve than Eq. (9-15).

To apply SCA to UD RVE, the first step is to build the UD RVE database.This involves two steps:

-   -   1. Compressed original RVE from voxel mesh into clusters.    -   2. Compute interaction tensor D^(IJ) between all cluster pairs.

Once the RVE is compressed, each voxel will be labeled with a cluster.This is illustrated in FIG. 159 where the RVE is decomposed into 10clusters: 2 in the fiber phase and 8 in the matrix phase.

Once the UD database is built, Eq. (9-16) is solved to compute stressand strain responses in each cluster when an external loading is given.This rom, hereinafter referred as udsca, can be used to computeelasto-plastic responses of UD RVE in a timely fashion. A numericalverification of UDSCA is performed as shown in FIG. 55, where transversetensile loading is considered. In this verification case, two differentROM resolutions are used: one with 16 clusters in the matrix phase andthe other with 8 clusters in the matrix phase. The number of clusters inthe fiber phase is kept as two. The result showed that using 8 clustersin the matrix phase and 2 clusters in the fiber phase provides goodaccuracy comparing to DNS solution. Hence, this ROM is used for all UDconcurrent modeling cases.

For UD 2-scale concurrent modeling, it follows the schematic shown inFIG. 160. The macroscale part is discretized as finite element mesh. TheROM of UD RVE intakes strain at the integration point and then passesback stress response to the integration point.

UD Off-Axial Coupon Tensile Concurrent Modeling

Next, UDSCA is applied to a coupon off-axial test model to performconcurrent multiscale modeling. For a realistic representation of theepoxy matrix, a paraboloid yielding surface is applied, where thetension and compression curves are extracted from FIG. 45.

Through the coupon test cases, two important problems are addressed:

-   -   (1) Computing material microstructure evolution on-the-fly by        realistic RVE.    -   (2) Prediction of CFRP part performance using the multiscale        method.

For the coupon model, exact geometry from NIST is used, as shown in FIG.161. The coupon model is built in commercial finite element softwareLS-DYNA. Note the teal region is the UD laminate made with 12 UDlaminae. It is impossible to model every single fiber in the couponexplicitly since at least 409,422 carbon fibers need to be modeled. If afinite element mesh shown before is coupled to individual integrationpoint of each finite element in the coupon mode, the computational costis still huge, and the estimated solution time is beyond the capabilityof existing HPC. However, using UDSCA, UD RVE responses at eachintegration point can be computed in an efficient manner.

The simulation took 2560 CPUS hrs to complete. The stress vs. straincurve in the y-direction is computed and compared with the experimentalresult, as shown in FIG. 1253. A summary of the coupon test is given inTable 9-11, where the concurrent model is able to predict ultimatestress and strain reasonably well.

FIG. 9-15. Normal Stress and Strain Curve of UD off-axial Coupon Test.

TABLE 9-11 Comparison of Ultimate Normal Stress and Normal Strain.Prediction Experiment Difference Ultimate Normal 404.809 MPa 395.639 MPa2.3% Stress Ultimate Normal 0.011 0.0118 6.7% Strain

In addition, FIG. 162 shows the von Mises stress of local RVEs and thecoupon before crack initiation and after crack formation. In FIG. 162,RVEs that represent four different integration points are visualized. InRVEs representing integration points on the crack, stress becomes zeroas the integration point is deleted from the coupon model. In RVEsrepresenting integration points that are not deleted, stress is stillnonzero due to stress wave propagation. The concurrent capture ofmacroscale and microscale stress evolution is made possible by theconcurrent multiscale modeling scheme.

UD Crash Test Concurrent Modeling

The UD crash test setup is shown in panel (a) of FIG. 163, where themodel is a quarter model based on the model provided by ford. The sameconcurrent scheme shown in ud coupon test is used.

The impactor force vs. Time is shown in panel (b) of FIG. 163 up to 4 s.It can be concluded that the time interval between the 1^(st) peak forceand the 2^(nd) peak force predicted by the concurrent model is similarto that of the test data. The peak force recorded is given in Table9-12, with a difference of 13.2%.

TABLE 9-12 Comparison of Peak Force Peak Force Prediction   11 × 1e4 kNTest 12.68 × 1e4 kN

UD Dynamic 3-Pt Bending Concurrent Modeling

The udsca is also applied to the ud hat-section 3-pt bending model. Themodel is shown in FIG. 164, along with the schematic of informationexchange between the model and lower level UD RVE.

After a total displacement of s, failure in UD laminae is observed. Peakload and peak impactor acceleration are reported in Table 9-13, wherethe difference is within 10% compared to test data.

TABLE 9-13 Comparison of Peak Load and Peak Impact Acceleration Peakimpactor Peak load(n) acceleration (m/s²) Experimental data 10328.10.390 Concurrent simulation 9660.0 0.382 Difference, % 6.47% 2.05%

Reduced Order Modeling of Woven CFRP

A 3-scale concurrent modeling for woven RVE has been established. Thescheme is illustrated in FIG. 166 below. In FIG. 165, stress response atthe macroscale integration point is computed by the ROM of woven RVEusing sca method. For each yarn cluster of woven RVE, a ROM of UD RVE isassigned and solved by SCA. Considering the information across threedifferent scales, this scheme is called 3-scale concurrent modeling.Note that there is no validation plan for 3-scale concurrent modeling inthe invention. Numerical samples are shown to illustrate the concept.

The geometry details of woven RVE is shown in FIG. 166 where allparameters are provided by cao group. In the woven RVE, the volumefraction of yarn is 77% and volume fraction of fiber in each yarn is60%.

A comparison between 2-scale and 3-scale single element simple sheartests is performed. Stress and strain results are shown in FIG. 167. Itis reasonable to conclude that 2-scale model predicts stiffer shearresponses of woven RVE since yarn phase is considered as elasticmaterials. The 3-scale model will require more computation cost due tothe consideration of UD RVE and it can be used when woven exhibitshighly nonlinear responses due to plasticity.

If the yarn nonlinearity (such as plastic behavior) is not of interest,it is also possible to replace UD RVE with a set of elastic constants ofUD with a fiber volume fraction of 60%. This will reduce 3-scaleconcurrent model to 2-scale concurrent model for woven, where the onlymatrix is modeled as elasto-plastic material. An orthogonal wovenbiaxial tension test shows almost linear stress and strain curves.Hence, a 2-scale concurrent modeling of woven bias tension simulation isperformed. The test setup is shown in FIG. 168. The same geometry isused in the concurrent modeling of the woven bias sample. Note here thewoven sample is made of single layer of woven microstructure shown inFIG. 166. The final σ₂₂ contour and σ₂₂ vs. ε₂₂ plots are given inpanels (a)-(b) of FIG. 169, respectively. In panel (b) of FIG. 169, theσ₂₂ vs. ε₂₂ are the averaged values obtained from those elements in thegauge zone shown in FIG. 168. The predicted τ₁₂ is slightly lower thantest data, but it shows the same trend. The difference might be due tovarious factors, such as inaccurate woven geometry and yarn properties.Further investigation is required.

Benefits Assessment

The UD and woven CFRP RVE modeling packages provide alternativesolutions to investigate CFRP mechanical properties. It can be appliedto different constituents and predict elastic stiffness tensors of UDand woven composite. The UQ function of woven RVE allows one to linkmanufacturing process parameters to final product performance.

Moreover, the UD and woven concurrent multiscale modeling provides anaccurate and efficient prediction of part-level product performance. Itcan be applied to a virtual verification platform where the conceptdesign is evaluated. It allows optimization of the new design and cansignificantly reduce the number of costly experiments. The potentialclients of this technology can be broad, as long as there is a need fordeveloping new composite applications.

In the multiscale modeling work, it is understood that microstructureplays an important role in modeling CFRP materials. An efficient reducedorder modeling method sca is introduced to integrate CFRP microstructure(UD and woven) into the part-level model to predict structuralperformance.

The next step will involve investigation of modeling of UD dynamicproblems, such as ud hat-section crash and dynamic 3-pt bending. Currentmodels suffer from numerical instability due to high loading rate. It isexpected to use different stabilization methods to improve theconcurrent scheme for better accuracy.

4. Stochastic Multi-Scale Characterization

Our research enables investigating the variability of part propertiesand behavior as a function of uncertainty sources at multiplelength-scale and, subsequently, identifying the most importantuncertainty sources that should be monitored during manufacturing. Thedeveloped methods and tools enable modeling spatiotemporally withvarying uncertainty sources and, additionally, couple structural andmaterial-related uncertainties across different length-scales. Weachieve these by introducing the Top-down sampling method that buildsnested multi-response Gaussian processes to parsimoniously quantify therandom fields and, hence, the underlying physical uncertainty sources.Our approaches can be easily used to conduct sensitivity analyses fordimensionality reduction, i.e., identifying the most importantuncertainty sources as well.

Compared to prior model-based UQ research, the UQ study of UD compositesin the invention is image-based and microstructure-oriented. Two sourcesof uncertainty are considered: fiber waviness and fiber spatialdistributions, both of which can be characterized from microscopicimages provided by Ford. Machine learning and applied statistics methodsare utilized to develop image analysis tools to extract informationabout the variations of the uncertainty sources, and generativestatistical models are constructed for generating realistic randomsamples with variations mimicking our observations from the image data.For fiber spatial distribution, tree regression is used for imagecharacterization and a hierarchical nonparametric sampling method isdeveloped to sample the realizations from a nonstationary andnonhomogeneous RF. The local fiber waviness is obtained from imagesthrough a specially designed segmented regression algorithm, and newrandom samples are generated via a frequency domain time series analysisapproach. Finally, the joint sampling of the two quantities, in whichspatial constraints exist, are discussed and the corresponding codes areimplemented.

The developed computational method and tools are applicable to manymaterial systems and the corresponding multiscale material simulators.We have demonstrated their effectiveness in modeling uncertainty sourcesin unidirectional and cured woven composites. Our computational methodsand tools can be validated against experimental results once they areavailable. In the case with cured woven composites, we demonstrate howvarious uncertainty sources such as yarn angle and fiber misalignment,which are introduced at, respectively, mesoscale and microscale, canaffect the part performance during operational conditions. Our resultsindicate that, even in linear analyses, such uncertainty sources couldhave significant impacts on the results. With the UD UQ tools, randomsamples that represent the variations in the real UD material can begenerated for further computational study, including their impact onmaterial properties and part performance.

Our contributions are the first to investigate the uncertainty inmultiscale material simulations which allows the systematic study of theeffect of uncertainty to (i) engineer more reliable materials, and (ii)reduce the manufacturing costs by only monitoring the main uncertaintysources. Composite vehicle components can be optimized considering theimpact of uncertainty, yielding a safer yet lighter design.

The Gaussian Process modeling tool developed during the project has beengeneralized into a user-friendly graphical user interface. The advantageof this tool is twofold; (i) The simplicity yet complimentary userinterface allows engineering teams company-wide to benefit from thispowerful tool, and (ii) Gaussian process models can reduce the effectivesimulation turn-around time from days to seconds, enabling the use ofthese models for uncertainty quantification and propagation purposes aswell as design optimization.

Uncertainty sources are generally categorized as aleatory and epistemic.While the former uncertainty source is inherent to the system (and henceirreducible), the latter is generally due to lack of knowledge or data,and may be reduced by conducting more simulations, experiments, orin-depth studies. In the case of materials, both sources are present andmay be introduced in the design and constituent selection stages,manufacturing processes, or operation. Such uncertainties manifest as,e.g., mechanical (e.g., Young's modulus, Poisson ratio, yield stress,damage evolution parameters, etc.) or geometrical (e.g., reinforcementdistribution, fiber misalignment in composites) variations. To elaboratemore on material uncertainty, we take woven fiber composites as anexample. These materials have been increasingly used in aerospace,construction, and transportation industries because of their superiorproperties such as high strength-to-weight ratio, non-corrosivebehavior, enhanced dimensional stability, and high impact resistance.Woven fiber composites possess, as illustrated in FIG. 171, ahierarchical structure that spans multiple length-scales from nanoscaleto macroscale. Within each of these length-scales, many correlated andspatially varying uncertainty sources are introduced: The high pressureand flow of the resin or draping change the local architecture of thefibers during the preforming process. Additionally, processingvariations and material imperfections cause the fiber volume fraction tospatially vary across the sample. These variations are particularlypronounced along the yarn paths where there is compact contact. Thesemacroscopic uncertainties are manifestations of many uncertainty sourcesthat exist at the finer scales where the number and dimensionality ofthe uncertainty sources increase due to the delicacy of materials.

FIG. 170 shows a multiscale structure: Schematic view of a four-scalewoven fiber composite with polymer matrix. In computational modeling ofthis structure, each integration point at any scale is a realization ofa structure at the next finer scale.

Fibrous composites have been previously investigated to determine howmuch their properties and performance are sensitive to uncertainty. Thefocus of these works, however, has not been placed on rigorouslymodeling the uncertainty sources and statistically propagating theireffects across multiple scales. For instance, modeling spatialvariations via RFs, connecting them across different spatial scales, andinvestigating stochastic simulations are often neglected. Savvas et al.studied the necessary RVE size as a function of spatial variations offiber volume fraction and yarn architecture. Their research showed thatthe RVE size should increase at higher fiber volume fractions. They alsoconcluded that the mesoscale RVE size is more affected by fiberorientation than waviness. Their further studies illustrated thatgeometrical characteristics (i.e., the shape and arrangement of thefibers) and the material properties (Young's moduli of the constituents)affect the homogenized response in UD composites quite significantly(with the former being more important). The variations were shown todecrease as the number of fibers and RVE size increased. Average axialand shear stiffness constituted the response in these studies.Vanaerschot et al. studied the variability in composite materials'properties and concluded that the stiffness in the mesoscale RVE isaffected by the load orientation and, additionally, it significantlydecreases as the fiber misalignment increases. Hsiao and Danielexperimentally and theoretically investigated the effect of fiberwaviness in UD composites and demonstrated that it decreases composite'sstiffness and strength under uniaxial compression loading. Komeili andMilani devised a two-level factorial design at the mesoscale to studythe sensitivity of orthogonal woven fabrics to the material propertiesand yarn geometry. They illustrated that, based on the applied load,these parameters could have a significant effect on the global response(i.e., reaction force). A similar sensitivity study based on Sobol'sindices was conducted in to demonstrate that the friction coefficientand yarn height significantly affect the macroscale mechanical responseof interest in dry woven fabrics. Yarn properties are spatiallyhomogeneous and there is no fiber misalignment.

To address the shortcomings of the prior works on UQ in wovencomposites, we employ RFs which are collections of random variablesindexed in either time or space. We introduce the Top-down samplingmethod that builds nested RFs by treating the hyperparameters of one RFas the responses of another RF. This nested structure allows us to modelnon-stationary and C⁰ (i.e., continuous but not differentiable) RFs atfine length-scales (i.e., mesoscale and microscale) with a stationaryand differentiable RF at the macroscale. We motivate the use ofmulti-response Gaussian processes (MRGPs) to parsimoniously quantify theRFs and conduct sensitivity analyses for dimensionality reduction. Theresulting approach is non-intrusive (in that the computational modelsneed not be adapted to account for the uncertainties) and can leveragestatistical techniques (such as metamodeling and dimensionalityreduction) to address the considerable computational costs of multiscalesimulations.

For UQ of UD composites, we focus on two microstructural uncertaintysources that can be captured by imaging techniques: fiber distributionand fiber waviness. Microstructure images of UD plates are taken atFord, from which we measure these two quantities of interest (QoI) thenmodel their variation with statistical methods. We address the challengeof simulating a nonstationary and nonhomogeneous RF for fiberdistribution modeling by introducing a hierarchical nonparametricstatistical model. For fiber waviness, the image data of which isextremely limited, assumptions such as stationarity are made and a timeseries approach is applied to generate realistic samples from theunderlying distribution. The two methods are integrated into adata-driven sampling algorithm that can simulate the spatialdistributions of the two QoIs simultaneously for further computationalmechanics analysis.

The developed methods and tools are applied to fiber composites whichhave been increasingly used in aerospace, construction, andtransportation industries due to their superior performance. Ourcontributions, hence, have far reaching impacts on various sectors ofthe economy.

Multiscale UQ and UP with Application to Cured Woven Composites

Our approach for multiscale UQ and UP has two main stages: Intra-scaleUQ and inter-scale UP. We start by identifying the uncertainty sourcesat each scale and modeling them via RFs where one RF is associated witheach structure realization. We employ RFs with sensible (i.e.,physically interpretable) parameters for three main reasons: (i) Tocouple uncertainty sources across length-scales and enable theirpropagation from lower to higher scales, (ii) To connect the mostimportant parameters of the RFs to the features of the material systemand hence identify the dominant uncertainty sources in a physicallymeaningful way, and (iii) To allow for a non-intrusive UQ procedure bydirectly using the RFs' outputs in the multiscale FE simulations(instead of adapting the FE formulations for UQ and UP). Due to thesereasons, we use the best linear unbiased predictor (BLUP) representationof multi-response Gaussian processes. MRGPs enable sensiblecharacterization of uncertainty sources, are flexible andcomputationally efficient, and can be easily trained via available data.

At this point, the dimensionality in the UQ and UP tasks has beenreduced from the number of degrees of freedom in the multiscalesimulation to the few hyperparameters of the MRGP at the coarsest scale.However, depending on the material system and quantities of interest,generally not all the hyperparameters need to be considered in the UPprocess. Hence, further dimensionality reduction can be achieved byidentifying the dominant uncertainty sources and, equivalently, thecorresponding RF parameters through, e.g., sensitivity analysis.

The second stage of our approach starts by replacing the nestedsimulations at fine scales via inexpensive but accurate metamodels (akasurrogates or emulators) to decrease the computational costs of a singlemultiscale simulation from hours (or even days) to minutes. The choiceof the metamodel, its inputs, and its outputs depend on the nature ofthe FE simulation. Finally, the uncertainty at the highest scale isquantified by propagating the uncertainty from all the finer scales inthe UP process. During UP, various multiscale simulations are conductedwhere for each simulation one realization of the spatially varyingquantities are used in the multiscale material. To generate each ofthese realizations, we introduce the Top-down sampling approach whererealizations are assigned to the spatially varying parameters from thecoarsest scale to the finest scale in the material system. This samplingstrategy enables modeling (i) non-stationary and C⁰ (i.e., continuousbut not differentiable) spatial variations at the fine scales, and (ii)partial correlations between the various uncertainty sources within andacross scales. Although the top-down sampling method can be integratedwith any analytical RF, we have chosen MRGPs since they are sufficientlyflexible and possess a few hyperparameters which are all physicallyinterpretable. Additionally, other RFs can sometimes be converted intoGPs upon appropriate transformations. Our approach is demonstrated for acomposite with two length-scales in FIG. 172.

FIG. 171 shows demonstration of our approach for s two-scale structure:Spatial random processes (SRPs) are employed for generating spatialvariations which are connected through the top-down sampling procedure.

Multi-Response Gaussian Processes for Uncertainty Quantification

MRGPs are widely popular in RF and surrogate modeling and have been usedin a wide range of applications including UQ, machine learning,sensitivity analyses of complex computer models, Bayesian optimization,and trac Table 9-Bayesian calibration. For an RF with q outputs y=[y₁,y_(q)]^(T) and the field (e.g., spatial or temporal) inputs x=[x₁, ,x_(d)]^(T), the BLUP representation of an MRGP with constant prior meansreads as:

yabout

_(q)(β,c(x,x′)),  (15-17)

where

_(q) represents a q-dimensional Gaussian process, β=[β₁, , β_(q)]^(T) isthe vector of responses' means, and c(x,x′) is a parametric functionthat measures the covariance between the responses at x and x′. Onecommon choice for c(x,x′) is:

c(x,x′)=Σ⊗ exp{Σ_(i=1) ^(d)−10^(ω) ^(i) (x _(i) −x′ _(i))²}=Σ⊗r(x,x′),  (15-18)

where Σ is a q×q symmetric positive definite matrix that captures themarginal variances and the covariances between the outputs, d is thedimensionality of the field, ω=[ω₁, , ω_(d)]^(T) are the so-calledroughness or scale parameters that control the smoothness of the RF, and⊗ is the kronecker product. Note that the dimension of β and Σ dependson q, while that of ω depends on d. The parameters β, Σ, and ω arecalled the hyperparameters of an MRGP model and, collectively, enable itto model a wide range of random processes:

-   -   The mean values of the responses over the entire input space are        governed by β.    -   The general correlation between the responses (i.e., y_(i) and        y_(j), i≠j) over the input space is captured by the off-diagonal        elements of Σ.    -   The variations around the mean for each of the responses are        controlled by the diagonal elements of Σ.    -   The smooth/rapid changes of the responses across the input space        are controlled by ω.

In case some experimental data are available, all the hyperparameters ofan MRGP model can be estimated via, e.g., the maximum likelihood method.Otherwise, as in this work, expert or prior knowledge can be used toadjust these parameters and model a spatially varying quantity. Oncethese hyperparameters are determined, generating realizations from anMRGP model is achieved through a closed-form formula.

Top-Down Sampling for Uncertainty Propagation

To carry out one multiscale simulation, material properties must beassigned to all the IPs at all scales where the IP properties at anyscale depend on an RVE at the next finer scale (this RVE itself has manyIPs). Since these properties depend on the uncertainty sources (or,equivalently, on the RFs), the latter must be coupled across the scales.Recall that, due to the multiscale nature of the structure, the numberof RFs significantly increases at the fine scales because we associatean RF to each structure realization.

Having used RFs whose parameters are physically sensible and can bedirectly linked to the uncertainty sources, this cross-scale coupling isstraightforward and can be achieved with top-down sampling where theoutputs of the MRGP at each IP of a particular scale serve as thehyperparameters of the MRGP of the RVE associated with that IP. Thisprocess constitutes nested RFs. To assign values to the IP parameters inthe entire multiscale structure, this approach starts from the coarsestor top scale and hence the name top-down sampling.

While the Top-down sampling method works with any parametric RFrepresentation (e.g., PCE or KL expansion), we highly recommendemploying compact representations that include a few hyperparameters.This is mainly because the number of hyperparameters at the coarsescales increases rapidly as the number of spatially varying quantitiesincreases at the fine scales. For instance, assuming three (two)quantities change spatially in a 3D microstructure, an MRGP with 12 (8)hyperparameters is required. To model the spatial variations of these 12(8) hyperparameters at the mesoscale, an MRGP with 93 (47)hyperparameters is required.

Case Study on Cured Woven Fiber Composites

We now follow the steps of our approach to quantify the macroscaleuncertainty in the elastic response of a cured woven composite as afunction of spatial variations in seven uncertainty sources: fibervolume fraction and misalignment, matrix and fiber modulus, and yarns'geometry parameters (i.e., yarn angle, height, and spacing). Asillustrated in panel (a) of FIG. 172, the structure is composed of fouridentical woven plies that are stacked in the same orientation andconstitute a total thickness of 2.4 mm (the fiber orientations areindicated with light blue lines).

FIG. 172 shows the macroscopic cured woven laminate structure. Panel (a)shows the deformed structure. The light blue lines indicate the fiberorientation. The dimensions are scaled for a clearer representation.Panel (b) shows the deterministic spatial variations of yarn angleobtained from simulating a perfectly manufactured composite. Panel (c)shows Von Mises stress field corresponding to Case 9. Panel (d) showsthe random spatial variations of yarn angle corresponding to one of therealizations of Case 1. Panel (e) shows the random spatial variations ofθ ¹ corresponding to one of the realizations of Case 3.

The geometry of these woven plies is obtained via the bias-extensionsimulation of woven prepregs using the non-orthogonal constitutivepreforming model. While the bottom of the sample is clamped, the otherend is pulled by 1 mm to generate the bias tension deformation. In themacroscale simulations, 3D solid continuum elements are employed todiscretize the structure. As our focus is on UQ and UP, at this point wehave assumed that only elastic deformation occurs.

Uncertainty Sources

Longitudinal fiber and matrix moduli, E_(f) and E_(m), are the first twouncertainty sources. Given the moduli, the yarn material propertiesprimarily depend on two parameters: fiber volume fraction (in yarn), v,and fiber misalignment. While in most previous works v is postulated tobe spatially constant, in practice, it varies along the yarn pathparticularly where yarns have compact contact. Consequently, our nextuncertainty source arises from the spatial variations of v which startsfrom the microscale and propagates to mesoscale and macroscale. In thiswork, we have assumed that 45%≤v≤65% based on our material system.

During the manufacturing process, the fibers in the yarn deviate fromthe ideal orientation and render the cross-section of the yarnheterogeneous and anisotropic. These deviations result in fibermisalignment which is different from the concept of fiber waviness inthat a fiber can be perfectly waved without misalignment. As illustratedin FIG. 173, this misalignment can be characterized by the two angles ϕand θ which measure the deviation of the fiber direction, ƒ₁, withrespect to the local orthogonal frame on the yarn cross-section, g_(k).Based on the available experimental data in the literature, in this workwe have assumed −π≤φ≤π and 0°≤θ≤90°.

In modeling the mesoscale woven composites, the yarn architecture isoften presumed to be perfect where the yarn angle, α, is set to 90° andthe yarn height, h, and spacing, s, are fixed to their nominal values.These assumptions do not hold in practice due to the large in-planeshear deformation during preforming process and manufacturingimperfections. Hence, we also investigate the effect of the spatialvariations of the woven RVE architecture (α, h, and s) on themacroscopic properties.

FIG. 173 shows fiber misalignment angles. The zenith and azimuth anglescharacterize the fiber misalignment angle with respect to the localorthogonal frame on the yarn cross-section.

Lastly, we note that in our example the deterministic spatial variationof α in a perfectly manufactured composite is, as opposed to the otherparameters (i.e., [v, φ, θ, h, s]), available from the preformingprocess simulation. This deterministic variation is used as the priormean (β in Eq. (9-17)) of spatial distribution of α while for the othersix parameters the nominal values are employed as the (spatiallyconstant) prior mean. In all seven parameters, the posterior spatialvariations are stochastic.

We employ the computational homogenization technique for modeling themultiscale woven sample where the material property at any length-scaleis calculated through the homogenization of an RVE at the lower scale.At the microscale, the RVEs include 300×300×60 voxels (42 μm×42 μm×8.4μm) and the fibers have a diameter of 7 μm. The simulations are elasticwhere periodic boundary conditions (PBCs) are employed. It is assumedthat the fibers and the matrix are well bonded and there are no voids.To obtain the stiffness matrix, C, of the UD RVE, six stress-freeloading states are applied (i.e., only one of the ε_(xx), ε_(yy),s_(zz), ε_(xy), ε_(xz), and ε_(yz) strain components are applied in eachcase). Since the simulations are elastic, C mainly depends on the volumefraction, v.

At the mesoscale, the open source software TexGen is used to create thegeometry and mesh for the 2×2 twill woven RVE with 8 yarns. The spacebetween the yarns is filled with matrix and voxel meshes are used todiscretize the RVE where each voxel is designated to either a yarn orthe matrix. To balance cost and accuracy, we have used a voxel mesh with625000 elements. To reduce the computation costs, PBCs are employedthroughout.

The nominal properties of carbon fibers and epoxy resin were taken frommanufacturer's data (see Table 9-14). The resin is isotropic, and itsmaterial properties are taken from pure epoxy. Yarns with well-alignedfibers are treated as transversely isotropic. With fiber misalignment,however, yarns are not transversely isotropic since the material frameacross the IPs on their cross-section is non-uniformly distributed. Inthis case, the micro-plane triad model is employed to account for fibermisalignment by defining an orthotropic micro-triad, {right arrow over(ƒ)}_(k), for each IP of the yarn. This triad is related to the localframe, {right arrow over (g)}_(k) (see FIG. 174), via the misalignmentangles:

$\begin{matrix}{{\overset{\rightarrow}{f_{1}} = {{{\cos(\theta)}\frac{{\overset{\rightarrow}{g}}_{1}}{\left| {\overset{\rightarrow}{g}}_{1} \right|}} + {{\sin(\theta)}{\cos(\varphi)}\frac{{\overset{\rightarrow}{g}}_{2}}{\left| {\overset{\rightarrow}{g}}_{2} \right|}} + {{\sin(\theta)}{\sin(\varphi)}\frac{{\overset{\rightarrow}{g}}_{3}}{\left| {\overset{\rightarrow}{g}}_{3} \right|}}}},} & \left( \text{9-19)} \right. \\{{\overset{\rightarrow}{f_{2}} = {{{- {\sin(\theta)}}\frac{{\overset{\rightarrow}{g}}_{1}}{\left| {\overset{\rightarrow}{g}}_{1} \right|}} + {{\cos(\theta)}{\cos(\varphi)}\frac{{\overset{\rightarrow}{g}}_{2}}{\left| {\overset{\rightharpoonup}{g}}_{2} \right|}} + {{\cos(\theta)}{\sin(\varphi)}\frac{{\overset{\rightarrow}{g}}_{3}}{\left| {\overset{\rightarrow}{g}}_{3} \right|}}}},} & \left( \text{15-20)} \right. \\{{\overset{\rightarrow}{f_{3}} = {\overset{\rightarrow}{f_{1}}\hat{}\overset{\rightarrow}{f_{2}}}},} & \left( \text{15-21)} \right.\end{matrix}$

where |⋅| and {circumflex over ( )} denote, respectively, the norm of avector and the cross product. As for the local frame {right arrow over(g)}_(k), it is automatically generated by TexGen for each IP (eachvoxel at the mesoscale) once the woven RVE is discretized. We note that,the stiffness matrix at each yarn material point is obtained via theUD-RVE homogenization.

To link the mesoscale and macroscale, the stress-strain relations foreffective elastic material properties of woven RVE are required. Thisrelation can be written in terms of the symmetric mesoscale stiffnessmatrix as:

$\begin{matrix}{{\begin{bmatrix}\sigma_{11} \\\sigma_{22} \\\sigma_{33} \\\sigma_{12} \\\sigma_{13} \\\sigma_{23}\end{bmatrix} = {\begin{bmatrix}C_{11} & C_{12} & C_{13} & C_{14} & 0 & 0 \\\; & C_{22} & C_{23} & C_{24} & 0 & 0 \\\; & \; & C_{33} & C_{34} & 0 & 0 \\\; & \; & \; & C_{44} & 0 & 0 \\\; & {{Sym}.} & \; & \; & C_{55} & C_{56} \\\; & \; & \; & \; & \; & C_{66}\end{bmatrix}\begin{bmatrix}ɛ_{11} \\ɛ_{22} \\ɛ_{33} \\{2ɛ_{12}} \\{2ɛ_{13}} \\{2ɛ_{23}}\end{bmatrix}}},} & \text{(15-22)}\end{matrix}$

TABLE 9-14 Fiber and matrix properties: The moduli (i.e., E and G) areall in GPa. Poisson's ratios along different directions are alsoprovided. E_(zz) E_(xx) = E_(yy) v_(zx) = v_(zy) v_(xy) G_(xz) = G_(yz)G_(xy) Carbon 275 19.8 0.28 0.32 29.2 5.92 fiber Epoxy 3.25 3.79 0.390.39 1.36 1.36 resin

Top-Down Sampling, Coupling, and Random Field Modeling of UncertaintySources

To help clarify the descriptions, we first introduce some notation. Wedenote the three scales with numbers: 1→Macro, 2→Meso,3→Micro.Superscript and subscripts denote, respectively, scales and IPs.Variables with a bar represent averaged quantities over all the IPs at aparticular scale. For instance, v_(i) ¹ denotes the fiber volumefraction assigned to the i^(th) IP at the macroscale.

${\overset{¯}{\theta}}^{2} = {\frac{1}{N}\Sigma_{i = 1}^{N}\theta_{i}^{2}}$

represents the average misalignment (zenith) angle at the mesoscale fora woven RVE.

The uncertainty sources in our composite are summarized in Table 9-15.While some sources are only defined among different structures (underspatial variations across realizations), others possess an extra degreeof variation in that they also change within structures.

TABLE 9-15 Uncertainty sources for one macro structure realization: Thesources are fiber misalignment angles (θ and φ), yarn spacing and height(s and h), fiber and matrix moduli (E_(f) and E_(m)), fiber volumefraction (v), and yarn angle (α). Scale Uncertainty Sources n Micro v ³,E_(f), E_(m) None Meso θ_(i) ², φ_(i) ², v_(i) ², E_(f), E_(m), α², s²,h² θ_(i) ², φ_(i) ², v_(i) ² Macro NA θ_(i) ¹, φ_(i) ¹, v_(i) ¹, E_(f)_(i) ¹, E_(m) _(i) ¹, α_(i) ¹, s_(i) ¹, h_(i) ¹

Assuming the eight tows in a woven RVE are statistically independent andthe spatial variations within them originate from the same underlyingrandom process, a total of 12 hyperparameters are required to completelycharacterize the spatial variations of [θ_(i) ², φ_(i) ², v_(i) ²] by anMRGP (see Eq. (9-17)): three mean values (β=[β_(v), β_(φ), β_(θ)]^(T)),six variance/covariance values for Σ([σ_(vv) ², σ_(φφ) ², σ_(θθ) ²,σ_(vφ) ², σ_(vθ) ², σ_(φθ) ²]), and three roughness parameters(ω=[ω_(x), ω_(y), φ_(z)]^(T) where xyz denotes the cartesian coordinatesystem at the mesoscale). Once these parameters are specified, thespatial coordinates of the IPs in a woven RVE can be used to assignrealizations of v, φ, and θ to them. For each IP at the macroscale,however, these 12 hyperparameters serve as some of the responses of themacroscale MRGP whose other responses correspond to [E_(f) _(i) ², E_(m)_(i) ², α_(i) ², s_(i) ², h_(i) ¹]. Therefore, the macroscale MRGP has atotal of 173 hyperparameters (17 mean values for β, 153 uniquecovariance/variance values for Σ, and 3 values for ω=[ω_(η), ω_(ξ),ω_(ζ)]^(T) where ηξζ denotes the cartesian coordinate system at themacroscale). In the top-down sampling approach, first the 173hyperparameters of the macroscopic MRGP are prescribed. Then, this MRGPis sampled to assign 17 values to each macroscopic IP where 12 of themserve as the hyperparameters of the mesoscopic MRGPs, 3 of them areemployed to construct the woven RVE, and 2 of them to determine thefiber and matrix moduli. This process is illustrated in FIG. 174 where,for clarity, only two (out of the 17) hyperparameters at the mesoscaleare presented. FIG. 174 shows coupling the uncertainty sources acrossthe scales: The spatial variations of v and θ at the macroscale areconnected to those at the finer scales. For brevity, the coupling isillustrated only for the average values for the two quantities (i.e.,the mean of the RFs: β=[β_(v), β_(θ)]=[v ²,θ ²]).

Dimension Reduction at the Mesoscale Via Sensitivity Analysis

By modeling the spatial variations via RFs, the dimensionality of the UQand UP problem has decreased to the number of RF hyperparameters.Although this is a significant reduction, the considerable cost ofmultiscale simulations (even in the linear analysis) renders the UQ andUP process computationally demanding. To address this issue, we notethat depending on the property of interest a subset of uncertaintysources are generally the dominant ones in physical systems. Since ourcomposite undergoes an elastic deformation, we expect a small subset ofthe uncertainty sources (i.e., RF hyperparameters) to be important.

We conducted multiscale sensitivity analyses to determine which of the12 hyperparameters of an MRGP model are the most important ones (andmust be considered in UP) based on their impact on mesoscale materialresponse. Our studies included changing one of the hyperparameters(while keeping the rest of them fixed) and conducting 20 simulations toaccount for the randomness. Then, if the variations in the homogenizedresponse (effective moduli) were negligible, the hyperparameter wasdeemed as insignificant and set to a constant thereafter. All thesimulations in this section were conducted on a woven RVE with α=90°.

TABLE 9-16 Case studies to determine the effect of fiber misalignmentand its spatial variations: The triplets in the description fieldscorrespond to (θ ², var(θ²), var(φ²)). Case ID Description 1 (1°, 1,20²) 2 (1°, 5, 20²) 3 (1°, 10, 20²) 4 (3°, 1, 20²) 5 (3°, 5, 20²) 6 (3°,10, 20²) 7 (5°, 1, 20²) 8 (5°, 7, 20²) 9 (5°, 10, 20²) 10 (1°, 1, 50²⁾11 (1°, 5, 50²) 12 (1°, 10, 50²) 13 (3°, 1, 50²) 14 (3°, 5, 50²) 15 (3°,10, 50²) 16 (5°, 1, 50²) 17 (5°, 7, 50²) 18 (5°, 10, 50²)

We found that the homogenized moduli are affected by neither the sixcovariance/variance values (i.e., [σ_(vv) ², σ_(φφ) ², σ_(θθ) ², σ_(vφ)², σ_(vθ) ², σ_(φθ) ²]) nor the three roughness parameters ω=[ω_(x),ω_(y), ω_(z)]^(T). In case of average values (i.e., β), the averagefiber volume fraction (v) and zenith angle (θ), as opposed to that ofthe azimuth angle, were found to considerably affect the homogenizedresponse of a woven RVE at the mesoscale. The effect of average valueson the effective moduli are summarized in FIG. 175. As illustrated inpanel (a) of FIG. 175, the moduli increase linearly as a function ofaverage fiber volume fraction. Comparing the first nine Cases in panel(b) of FIG. 175 with the last nine Cases, it can be concluded thatvar(φ²) insignificantly affects the moduli and poison ratios. panel (b)of FIG. 175 also indicates that the deviations from the referencesolution increase as θ ² increases (θ ² increases between the jumps, seeTable 9-16).

FIG. 175 shows effect of average values on the effective moduli of awoven RVE: panel (a) Effect of fiber volume fraction and, panel (b)Effect of misalignment. Each point on these figures indicates theaverage value over 20 simulations. The Case IDs in panel (b) are definedin Table 9-15. The reference solution refers to a case where there is nomisalignment.

It is noted that since we are interested in the elastic response of themultiscale composite in FIG. 173, the variations of effective moduliwere only considered in our sensitivity studies. As opposed to theeffective behavior, the local behavior (i.e., stress field) of the wovenRVE is quite sensitive to the spatial variations of both v and θ (butnot φ) and must be considered in nonlinear analysis.

Replacing Meso and Microscale Simulations Via Metamodels

To further reduce the multiscale UQ and UP costs, we employ metamodelsto replace the micro and mesoscale FE simulations corresponding to eachmacroscale IP. In particular, the metamodel captures the macroscalespatial variations of the stiffness matrix of the woven RVEs associatedwith the macroscale IP's as a function of yarn angle (α²), averagevolume fraction (v ²), yarn height (h) and spacing (s), averagemisalignment angle (θ ²), and fiber and matrix moduli (E_(f) and E_(m)).In machine learning parlance, the inputs and outputs of the metamodelare, respectively, [α², v ², s², h², θ ², E_(f), E_(m)] and thestiffness matrix C of a woven RVE. To fit the metamodel, we generatedsix training datasets of sizes 60, 70, , 110 with Sobol sequence forα²∈[45°, 90°], s²∈[2.2, 2.5] mm, h²∈[0.3, 0.34] mm, v ²∈[40%, 70%],E_(f) ²∈[150, 400] GPa, E_(m) ²∈[1.5, 5] GPa, and θ ²∈[0°, 6° ].Afterwards, we fitted an MRGP metamodel to each dataset. The accuracy ofeach model was then evaluated against a validation dataset with 30samples via:

$\begin{matrix}{{e = {100\sqrt{\frac{1}{30}{\Sigma_{i = 1}^{30}\left( {1 - \frac{{\overset{\hat{}}{y}}_{i}}{y_{i}}} \right)}^{2}}\%}},} & \text{(15-23)}\end{matrix}$

where ŷ=[ŷ₁, , ŷ₃₀] and y=[y₁, , y₃₀] are obtained from, respectively,the metamodel and FE simulations. The prediction error of each model isillustrated in FIG. 176 where it is evident that with roughly 100samples all the elements of C can be predicted with less than 6% error.

FIG. 176 shows prediction error as a function of the number of trainingsamples: As the number of training samples increases, the accuracy ofthe MRGP metamodel in predicting the elements of the stiffness matrix ofthe mesoscale woven RVE increases.

Graphical User Interface (GUI) for Optimal Sampling and Metamodeling

As metamodeling is a broadly applicable tool (also outside the field ofstochastic multiscale modeling), two user-friendly Graphical Interfaceshave been developed: Optimal Latin Hypercube Sampling (OLHS), andGaussian Process modeling. The ideology behind these tools is to befunctional and complete, while being intuitive enough for novice users.Furthermore, the graphical interfaces have been developed using MatlabGuide and can be run on any 64-bit computer under the windows operatingenvironment.

As shown in panel (a) of FIG. 178, OLHS Generation and visualizationpackage allows for the automatic generation of training data inputs thatspan the desired metamodel space as optimal and uniform as possible. Anynumber of input variables and number of samples can be specified. Asshown in panel (b) of FIG. 178, our package for GRP is applicable tomulti-dimensional and multi-response datasets and can automaticallyhandle noisy observations once enough training data is provided.Additionally, the interface includes features that allow the user toevaluate metamodel accuracy, perform prediction on unobserved inputs,and visualization that allows the user to conveniently investigateinput-output relations regardless of problem dimensionality.

Results on Macroscale Uncertainty

Multiple macro simulations are conducted where θ_(i) ¹, v_(i) ¹, E_(f)_(i) ¹, E_(m) _(i) ¹, α_(i) ¹, s_(i) ¹, and h_(i) ¹ change spatiallyacross the macroscale IPs. To quantify the importance of thesevariables' spatial variations on the macroscopic behavior, eight casesare considered. The spatial variations are changed with an MRGP at themacroscale in a controlled manner from one case to the next. Except forthe last case, 20 independent macroscale simulations are conducted foreach case to account for the randomness. In summary, a total of 161macroscale simulations are conducted. The last case study serves as thereference where there is no misalignment, α_(i) ¹ is determined via theprocessing simulation over the structure as illustrated in panel (b) ofFIG. 172, and all other parameters are set to their nominal values,i.e., v_(i) ¹=55%, E_(f) _(i) ¹=275 GPa, E_(m) _(i) ¹=3.25 GPa, s_(i)¹=2.35 mm, and h_(i) ¹=0.32 mm. In cases 1 through 7, the effect ofspatial variations is considered for one parameter at a time where theamount of variations with respect to the prior mean is controlled by thevariance of the macroscale MRGP. In these cases, we set the varianceassociated with the spatially varying parameter to, respectively, 9, 25,1, 0.05², 0.01², 40², and 1. These variances are large enough to capturerealistic spatial variations while small enough to ensure that therealized values are not outside the ranges where the mesoscale MRGPmetamodel is fitted. Sample spatial variations for case 1 and 3 aredemonstrated in panels (b)-(e) of FIG. 172.

Since in reality the uncertainty sources coexist, in case 8 we considerthe effect of all the uncertainty sources and their correlation. Here,the individual variances (diagonals of macroscale MRGP, Σ) are the sameas in cases 1 through 7 while the covariances are chosen to reflect thenegative correlation between the fiber volume fraction and both yarn andfiber misalignment. To this end, we choose [σ_(αv), σ_(αθ), σ_(vθ)]=[−9,3, −3] and set the rest of the off-diagonals of Σ to zero. We note thatσ_(αv) is negative to model the increase in fiber volume fraction as theyarns get closer after preforming. σ_(vθ) is also negative to considerthe decrease in misalignment angle in rich fiber regions.

TABLE 9-17 Description of the simulation settings: Nine simulation casesare considered to quantify the macroscale uncertainties and the relativeimportance of spatial variations. MRGPs are employed to generate randomrealizations in all cases. Except for case 9 (reference simulation), allcases include 20 simulations to account for randomness where only oneuncertainty source exists. Description Case 1 Spatial variations in αwith its prior spatial distribution determined by processing simulationCase 2 Spatial variations in fiber volume fraction with a constant priorof 55% over the structure Case 3 Spatial variations in fibermisalignment with a constant prior of 3° over the structure Case 4Spatial variations in yarn spacing with a constant prior of 2.35 mm overthe structure Case 5 Spatial variations in yarn height with a constantprior of 0.32 mm over the structure Case 6 Spatial variations in fibermodulus with a constant prior of 275 GPa over the structure Case 7Spatial variations in matrix modulus with a constant prior of 3.25 GPaover the structure Case 8 Spatial variations in all parameters Case 9All parameters set to nominal values (α from processing simulation) andno misalignment

Panel (a) of FIG. 178 compares the exerted force on the cross-section ofthe sample for the nine cases where, for cases 1 through 8, theforce-displacement line is averaged over the 20 multiscale simulations.The results suggest that in the presence of fiber misalignment, thestructure weakens and hence the reaction force decreases. This weakeningis exacerbated in the presence of other uncertainty sources. Inparticular, while in the reference case the maximum reaction force(magnitude) is 15.4 kN, in cases 3 and 8 it is 14 kN and 13.5 kN,respectively. These results indicate that: (i) Fiber misalignment mustbe accounted for in the simulation of composites even if one isprimarily interested in the global response in linear analyses. (ii) Theinteraction among various uncertainty sources should be considered: onceall the uncertainty sources are accounted for, the structure is morenoticeably weakened with a 12.3% reduction in reaction force. (iii) Theinsensitivity to the spatial variations of some parameters can beexplained by the fact that in linear analyses the global response ismainly affected by the averaged properties. Since we only introducespatial variations (we do not change the parameter averages over thestructure except for fiber misalignment), the results intuitively makesense.

To illustrate the effect of spatial variations on local behavior, wecompare the average and standard deviation of the von-Mises stress fieldover the mid-section of the structure in panels (b)-(c) of FIG. 178,respectively. We choose the mid-section over the entire structure foranalyzing the strain field since an explicit solver is employed in ourmultiscale FE simulations where artificial stress concentrations mightoccur at the boundaries of the structure. The curves in these twofigures are obtained by analyzing the 20 simulations corresponding toeach case. Similar to panel (a) of FIG. 179, panel (b) of FIG. 178demonstrates that the structure weakens in the presence of fibermisalignment as the mean stress over the mid-section is lower than thatof the reference structure (case 9). It is evident that the mostrealistic case where all the uncertainty sources are present results inthe weakest structure. More interestingly, this weakening is not uniformover the mid-section and is the largest at the middle where thereduction is roughly 12.3%.

The highest variations among the simulations of a specific case areobserved in case 8, where all the parameters change spatially andrelatively, see panel (c) of FIG. 178. Cases 2, 3, and 7 are next inline with cases 2 and 7 having more fluctuations among the IPs (notethat each point along the x-axis corresponds to an IP in themid-section). Finally, cases 4 and 5, which correspond to thesimulations where the yarn spacing and height change, have the leastamount of variations. This is expected since the stiffness matrix of awoven RVE (obtained via the mesoscale MRGP metamodel) is alsoinsignificantly sensitive to these parameters. We highlight, however,that we have considered a relatively small range of variations for thesetwo parameters. Their effect will be more prominent if these ranges areincreased.

Image-Based Microstructural UQ of UD Composites

The UQ work for UD composites aims at statistical modeling andreconstruction of the material microstructural features, including (1)non-uniform fiber spatial distribution and (2) fiber waviness. We buildour models based on microstructure images of UD plates taken at FordMotor Company. Several machine-learning- and applied-statistics-basedapproaches are developed for image characterization, informationretrieval and generative model building. For the spatial fiberdistribution, which exhibits both non-stationarity and non-homogeneity,we first model the data (image) via a tree-regression algorithm then ahierarchical nonparametric sampling approach is developed. The approachis completely data-driven, in the sense that no probability models areassumed and a part of a new sample is generated by resampling from thedata.

Fiber waviness is the local orientation of the fiber bundles relative tothe global direction of the fibers. Perfectly straight fibers have zerowaviness everywhere, however, the transverse images taken fromunidirectional fiber composite samples (FIG. 182) show that wavinessdoes exist. On the other hand, due to the limitation in image quality,only partial fibers are observed in terms of disconnected fragmentstherefore traditional computer vision algorithms for edge or objectdetection are not applicable in the invention. To conquer thischallenge, we developed a segmented regression algorithm that canestimate the local waviness angle via a linear-regression-like-,optimization-based approach. Then the angle distribution along the fiberlongitudinal direction are modeled with a time series statistical model,from which we can sample realizations from.

Fiber Distribution Modeling

By visually inspecting the distribution of fibers, which is representedby distribution of local volume fraction of fibers (panel (a) of FIG.179), the spatial correlation between the local fiber volume fractionsexhibits two characteristics. The first one is non-homogeneity, which isreflected by the layered structure along the vertical direction. Thesecond one is non-stationarity, which refers to the local curvature ofthe volume fraction patterns along the horizontal direction. Modelingboth features with traditional parametric statistical models willinvolve estimation of lots of parameters representing these features.Therefore, we use an alternative approach: along the vertical direction,the non-homogeneity, i.e., the shape of some wave-like functions, issampled from its empirical distribution through resampling from theimage dataset (with replacement); along the horizontal direction, thenon-stationarity is modeled by looking for a representative statistic ofthis feature and attempting to find the distribution of the statistics.

The first step to find such a statistic is to reduce the dimension ofeach sample image from approximately 450,000 pixels to a manageablesize. To compress the data in a sample, we used a regression treealgorithm, i.e., a sample is represented by a tree-structured field. Thelocations of the splitting lines (nodes of the tree) are found byminimizing the integrated relative error (IRE), and between theseparation lines the data are interpolated linearly. By setting theregression goal to IRE <5%, we normally obtain 200-300 nodes. Since thesplitting lines separate the most distinct areas, the locations of themcontain the information of non-stationarity: the denser the lines are,the more local curvatures the area include (panel (b) of FIG. 179). Notethat a dense collection of the lines means smaller inter-line distances.It follows that we can use the distribution of the inter-line distancesto characterize the non-stationarity.

The distribution, in terms of probability density estimates, of theinter-line distances are generated with all the samples. The first threeplots of FIG. 180 shows three of them. It can be observed that eachsample has different levels of non-stationarity, which might account forthe difference in the material performance. The distributions in thelast three plots are estimated by grouping inter-line distance data of10 randomly selected samples together. The similarity in shape of theprobability densities suggests the convergence of the distribution,i.e., the common distribution behind all samples.

This observation provides a way of generating random samples withsimilar levels of non-stationarity, which is sampling the locations ofthe splitting lines from the common distribution in panel (b) of FIG.180. The volume fractions along the vertical splitting lines are sampleddirectly from the empirical distribution estimated from the data.Therefore, a sampling algorithm for the fiber distribution/volumefraction can be constructed by (a) generating random splitting locationsfrom the common distribution of inter-line distances, then (b)resampling the volume fractions along the splitting lines from thedataset, and (c) the areas between the lines are filled by interpolatingthe neighboring lines. Two example reconstructions are demonstratedbelow (FIG. 181), which shows both randomness and high level ofsimilarity compared to the original samples (for example, in panel (a)of FIG. 180).

Fiber Waviness Characterization

Characterization of fiber waviness from the transverse images is not asstraightforward as the same task with fiber distribution, as the localcurvature in the image, in terms of local slopes or angles, cannot becalculated directly from the pixel information (e.g., binary orgrayscale values). Normally the characterization process would involvedetecting each of the fibers on the image and calculate the anglesaccordingly. However, this approach fails in the invention because thefibers in the images are not fully shown, i.e., only partial fibers areobserved, and some of the parts appear in just dots or small pieces, theorientation of which are not measurable individually (see, for example,the binary image segment in FIG. 182). It is possible to design filtersto remove them as noises, but the fact that they still have a generaltrend as a group indicates that statistical characterization is helpfulin preserving the information. Also note that linear regression is anideal tool to estimate slopes, therefore we developed a segmented linearregression approach (FIG. 182) to characterize the local curvature inthe images.

The idea behind this method is: the relationship between the local angleand local slope is given by β=tan(α), where β is the slope and α is theangle, and the local slope is estimated by the slope of the regressionline with the points on a locally binarized segment of the originalimage. The challenging part is to build a valid linear model for theregression: the classic simple linear model assumes the error isnormally distributed and the well-known least squares method is derivedbased on this assumption, which is not valid in our problem. We proposedour own regression algorithm customized for this case: under theassumption that the fiber pixel point and fiber locations are uniformlydistributed, assume the origin of the coordinates is put in the centerof the image segment, and a regression-through-the-origin meanprediction is given by the function y=βx, the estimate of the slope β isgiven by:

$\begin{matrix}{\hat{\beta} = {\underset{\beta}{argmax}{f\left( {e;\beta} \right)}}} & \text{(9-24)}\end{matrix}$

where e is the set of residuals with the regression line y=βx, ƒ:

→

(assume there are n points in total) is the mapping between theresiduals and the number of modes in the probability density estimationof the residuals. With this estimation, the trend of the fibers in thetransverse direction are correctly captured (see FIGS. 183 and 184 forillustration).

Fiber Waviness Modeling

The challenge associated with this task is the very limited number ofimages: effectively only one in total (FIG. 184). Therefore, assumptionsare made to validate our approach to build the statistical model andsampling algorithm: (1) the waviness contained in this image isrepresentative of the waviness distribution, (2) the wavinessdistribution is stationary, and (3) the impact of the waviness changealong the thickness/vertical direction is trivial, therefore only thefiber angle distribution along the horizontal direction is modeled.Based on (9-3), we can average the waviness angles along the thicknessdirection and obtain a 1D signal (FIG. 185).

The resulting signal has a wave shape with varying amplitudes andfrequencies. A natural characterization of such signal is theperiodogram. A periodogram is the estimate of the spectral density of asignal, which can be obtained by discrete Fourier transform of a time orspatial series. Under assumption (2), it can be shown that theperiodogram of a series converges in distribution to a sequence ofindependently and exponentially distributed random variables as thelength of the series increases. The sampling algorithm is thenconstructed by (a) obtaining the periodogram of the signal, (b)generating a random periodogram by sampling from independent exponentialdistributions with parameters given by (a), and (c) using inverseFourier transform to obtain the new waviness sample from (b). FIG. 185shows the comparison between the original sample and the reconstructionsin both spatial and frequency domain.

Joint Sampling of Two Uncertainty Sources

The two algorithms above, one for fiber waviness and one for fiberdistribution, are capable of modeling and sampling the respective QoIindividually. However, when they are integrated into a joint samplingprogram, spatial constraints must be taken into account. For example, ifwe want to generate the spatial distributions of the two QoIs for acoupon simulation model simultaneously, the pattern of the fiberdistribution at different cross-sections (i.e., cross-sections taken atdifferent locations in the fiber longitudinal direction) should besimilar but different, because the fibers are curved along thelongitudinal direction. Therefore, combining independent realizations ofthe two sampling codes will not represent the real situation. Inobservation of this phenomenon, we developed a joint sampling algorithmfor coupon models and realistic realizations can be generated from thismethod.

Contrary to prior work that rely on random variables, we employ randomfields to model the spatially varying uncertainty sources in multiscalematerials such as cured woven composites. We introduce the Top-downsampling method that builds nested random fields and, in turn, allows usto model non-stationary variations at fine length-scales (i.e.,mesoscale and microscale). We motivate the use of multi-responseGaussian processes to parsimoniously quantify the random fields andconduct sensitivity analyses for dimensionality reduction. The resultingapproach is non-intrusive and can leverage statistical techniques toaddress the considerable computational costs of multiscale simulations.

The computational demand of multiscale materials has been circumventedby the use of Gaussian random processes trained on a space fillingdesign. As computationally demanding simulations are ubiquitous amongengineering problems, the developed user-friendly GUIs enable moreengineers working on a wide variety of challenges to benefit from thesepowerful tools. The GUIs and the tools embedded in their source code areable to reduce effective simulation turn-around time from days toseconds, including high dimensional problems of up to 50 inputvariables.

Image characterization techniques are developed to quantify thevariations in UD uncertainty sources (fiber waviness and spatialdistribution) and the corresponding statistical models are introduced tostudy the variability and sample realizations of random fields or randomprocesses from the underlying distribution. Compared to existing workthat often pre-assumes some parametric model to represent theuncertainty, our methodology fully utilizes the availablemicrostructural image data and enables the systematic study of theuncertainties of UD composites in a real-world setting.

Uncertainty quantification involves the use of a wide range ofcomputational and statistical tools and the choice of the appropriatetool depends on the available information and resources. For example,for UQ of woven composites, in the absence of data, parametric RF modelslike Gaussian RFs are chosen and the associated top-down samplingapproach for multiscale analysis is developed to study the impact ofuncertainty; while for UD of UD composites, images are given, hence thesampling algorithm is designed to mimic the information contained in theimage data so that the random reconstructed samples are realistic.

The domain size in UQ is also very important. For instance, the chosenmicroscale and mesoscale RVEs in UQ of woven composites weresufficiently large and so more uncertainty would have been observed inthe response had we chosen smaller RVEs. Finally, we note that theunderlying assumptions of any method (including ours) can be validatedvia experimental data. For instance, our choice of MRGPs for UQ in wovencomposites implies a normality assumption which might not hold in otherapplications where the distributions are heavily skewed or appropriatetransformations are not readily available. In such cases, other randomfield representations can be integrated with the top-down samplingapproach at the expense of more computations

In the future, we plan to gather experimental images from variousstructures at different spatial scales to further demonstrate theapplications of our approach. Experimental data would additionally allowus to validate the normality assumption made on the marginal variations.Finally, since uncertainty is more important in nonlinear analyses ofmaterials and structures, we plan to apply our approach to nonlinearproblems such as plasticity.

For UQ of UD composites, after the variations in the uncertainty sourcesare quantified and corresponding sampling algorithms are developed, thenext step is to conduct FEA simulations based on the generated samplesand study the effect of the uncertainty sources. If they are proved tobe influential to the properties of the UD composites, other multiscalemodels that utilize the UD properties should be modified to reflect thepresence of uncertainties at the scale of UD. It will also be beneficialif some non-image-based, parametric models are developed for UQ of UD sothat the level of uncertainty can be controlled by some parameters tomake possible studies like sensitivity analysis and metamodeling.

5. Part-Level Molding and Model Validation

Part-scale preforming experimental validation in this section measuresthe prediction accuracy of the non-orthogonal prepreg material model andthe multiscale preforming simulation method. It also provides guidanceabout the limitations of current simulation methods and possiblesolutions in the future. Finally, selection of various processparameters in preforming validation experiments leads to different partqualities. Observation and summary of the relation between processparameters and parts' quality can serve as an empirical rule to producehigh-quality parts either for research or for real productionapplication.

With the introduction of double curvature features, the double-domevalidation experiment applied in this CIME project gives a trustworthyapproach to quantitatively measure prediction accuracy of preformingmodels for 3D geometry parts.

This double-dome preforming experimental validation can be used for notonly the models developed in the invention, but also future models forcarbon fiber prepreg preforming simulation to determine their accuracyand application potential.

Double-dome benchmark geometry is used to validate the non-orthogonalmaterial model and the multiscale preforming simulation method developedin the invention at part-scale. Double-dome geometry applied in theinvention has 3D shape and complex double curvature features at the sizeof common automobile parts. It is an ideal benchmark to quantitativelymeasure prediction accuracy of the preforming models for real partproduction. In validation, not only the final part shape, but also theyarn angle at different locations, and forming force, are compared. Thisis because yarn angle, which is an indicator of fiber orientation,significantly affected mechanical stiffness and strength of CFRP parts,while forming force is an indication of membrane stress that controlstow separation and breakage. Different process parameter combinationsare tested in this benchmark validation in order to ensure that themodels can work at various production conditions. Comparison between thenon-orthogonal model and conventional orthotropic material models isalso performed to show the improvement we achieved for preformingsimulation in the invention.

The major technical target for this part-scale double-dome preformingexperiment is to validate the prediction capability of thenon-orthogonal material model and the multiscale preforming simulationmethod we developed in the invention, and to check whether the fiberorientation prediction from these two simulation approaches achieves theproposed 5% error. With successful establishment of this benchmark testand corresponding quantitative measure and validation criteria, we alsoaim to provide a widely accepted preforming simulation validation methodfor both academic and industrial researchers.

Preforming is a temperature varying process because of the utilizationof hot prepreg sheets and cold/warm tools in the process. In thedouble-dome benchmark preforming validation experiments performed forthe invention, supplied prepregs were first heated in an oven to around70° C. and then placed in a press for preforming. The geometry of thedouble-dome punch and the binder are demonstrated in FIG. 186, while theexperimental setup is illustrated in FIG. 188. For fast production rate,the press was kept at 23° C. by the coolant running within it, thereforethe temperature of the prepregs dropped from the initial value duringpreforming. In a single layer preforming setup, temperature history atthe top surface center, bottom surface center, and one side point on thetop surface of one prepreg are measured by thermocouples and plotted inpanel (b) FIG. 186. The plot indicates that the prepreg reached atemperature of around 70° C. in the oven. Then, it was cooled downgradually to around 45° C. by air during the transportation from theoven to the press. When placed in the press, the cooling rate increasedgreatly due to heat conduction between the hot prepreg and cold metal.Specifically, the temperature dropped 20° C. within the first 2 seconds.Meanwhile, it took the press 10 seconds to contact the punch and theprepreg and another 6 seconds to finish preforming. Therefore, theactual temperature of the prepreg during this preforming process wasvery close to 23° C., i.e., the press temperature. As a result, althoughthe preforming models are capable of taking temperature-dependence intoaccount, in the current validation experimental setup, it is reasonableto characterize the mechanical properties of prepregs and simulate thepreforming process at a fixed temperature of 23° C.

Preforming simulation models, utilizing either experiment-basednon-orthogonal material model or multiscale method, were established inLS-DYNA® using the dynamic explicit integration method. The simulationsetup is illustrated FIG. 188: layers of prepregs are preformed in thisprocess where punch displacement is 90 mm, and binder force increaseslinearly from 4000 N to 8200 N from experimental measurements. Thicknessof the prepregs is orders of magnitude smaller compared to its lengthand width, so the prepregs are discretized by reduced integrated shellelement S4R to reduce the computational cost. Each element is about 4mm×4 mm with five through-thickness integration points. Tools aresimulated as rigid bodies because of their high stiffness compared tothe soft prepregs with uncured resin, hence, their element type will notaffect simulation results. S3 elements are selected to discretize thetools because of their excellent auto-mesh capability for complexgeometries. Friction coefficient between tools and prepregs is set to0.3 according to the experimental measurement. The friction coefficientis the constant dynamic one in LS-DYNA® and the static friction isneglected because the preforming process leads to large prepregdeformation which, in turn, results in large sliding between the toolsand prepregs.

For the first set of validation, only one layer of prepreg with ±45°initial fiber orientation was preformed. The results from theexperiment, simulation with experiment-based non-orthogonal materialmodel, and simulation with conventional orthotropic material model, arecompared. In simulation models, material properties were calibratedusing uniaxial tension, bias extension and bending tests at 23° C.,while the initial angle between the yarn direction and the globalcoordinates was defined as a material input property to identify thefiber layup. The simulation results in the upper-right quarter of FIG.125 shows that the non-orthogonal material model established is capableof accurately predicting the physical experiments regarding the yarnangle distribution and blank draw-ins. For instance, the deviation ofthe maximum draw-in distance is about 9 mm, as listed in Table 9-1. Forcomparison, an orthotropic material model (MAT_002) is utilized inanother simulation whose result is shown in the upper-left quarter ofFIG. 189. Since the orthotropic model cannot track material propertychanges during the rotation of yarns, the corresponding simulation has amaximum draw-in deviation of 24 mm, as listed in Table 9-1, notcapturing the overall process behavior.

In the non-orthogonal model, yarn angle is defined as an outputvariable, while MAT_002 does not have the capability for directvisualization. For clarity, Table 9-18 compares the resulting shearangles at various locations obtained from the experiment andsimulations. Again, it shows that the current model has improved theprediction accuracy. The fiber orientation (yarn angle) predictionerrors at only half of the locations reach the proposed 5%, which isunsatisfactory.

TABLE 9-18 Draw-in distance and yarn angle comparison between simulationand experiment: simulation results are from the experiment- basednon-orthogonal material model and the conventional orthotropic materialmodel (MAT_002). Comparison Draw-in A B C D E F Non-orthogonal 40.22 mm89° 89° 71° 40° 45° 65° Orthotropic 25.00 mm 70° 85° 86° 47° 59° 77°Experiment 49.02 mm 80° 88° 71° 49° 56° 66°

The reason for the unsatisfactory fiber orientation prediction accuracyis that this non-orthogonal material model only utilizes experimentaldata from uniaxial tension (pure tension) and bias-extension (pureshear) tests. The coupling between tension and shear is neglected. Forreal prepregs, an increase of tension along the yarns will increase thecontact force and friction force between warp and weft yarns, resultingin a shear resistance increase. This kind of mechanism is not simulatedby the experiment-based non-orthogonal model. It is, however, capturedby the multiscale preforming simulation method, where virtual materialcharacterization is performed by experimentally calibrated mesoscopicRVE models that can be deformed to arbitrary strain. To demonstrate theimprovement from multiscale modeling, its simulation result is comparedagainst the one obtained from the previous non-orthogonal materialmodel. Final prepreg geometry and yarn angle distribution results aredemonstrated in panel (a) of FIG. 190 together with the real preformedpart. The draw-in distance and the yarn angle at the sampling locationsfrom the simulation and the experiment are listed in Table 9-19. Thecomparison indicates that this multiscale method with tension-shearcoupling fulfils the proposed 5% error of fiber orientation (yarn angle)prediction at 5 out of 6 sampling points, and prediction errors at allsampling points are less than 8%.

TABLE 9-19 Draw-in distance and yarn angle comparison between thesimulation and the experiment: The simulation results are from the newmultiscale material model and the tension-shear decoupled materialmodel. Comparison Draw-in A B C D E F Multiscale 42.25 mm 86° 88° 73°54° 57° 67° Non-orthogonal 40.22 mm 89° 89° 71° 40° 45° 65° Experiment49.02 mm 80° 88° 71° 49° 56° 66°

The punch force-displacement curves from the two simulation cases andthe experiment are compared in panel (b) of FIG. 190. The plotsdemonstrate that the multiscale preforming simulation method predictsthe punch force nearly the same as the experimental one compared to thesimulation using the experiment-based non-orthogonal material model,which underestimates the experimental punch force by around 26%. Thesmall discrepancy between the forces from the new simulation method andthe experiment when the punch displacement reaches to over 70 mm may becaused by the negligence of the prepreg thickness variation by the shellelements in the simulation. The small force discrepancy when the punchdisplacement ranges from 20 to 50 mm, however, may result from the factthat the temperature at some locations of the prepreg has not completelyreached 23° C. at the initial stage of the preforming, leading to softermaterial behavior compared to the one for the simulation. As a summary,this multiscale preforming simulation method with tension-shear couplingcan predict the draw-in distance and the yarn angle variation on thepreformed prepreg. More importantly, it also predicts the punch forcehistory with high accuracy. This multiscale method, therefore, hasstronger predictive capability than the experiment-based non-orthogonalmode does, and can serve as a powerful tool for part performanceprediction, process parameters optimization, material design, and defectanalysis for future preforming works.

After single layer preforming validation with a ±45° initial fiberorientation and 6 yarn angle measuring locations, further validationwith different measuring locations, different initial fiberorientations, and different numbers of prepreg layers (0°/90° one layer,±45° one layer, and 0°/90°/±45° two layers) were performed. Because ofproject time limitations, only the simulation results from theexperiment-based non-orthogonal model are compared with the experimentalones. The multiscale simulation method was not applied to theseconfigurations.

Simulation and experiment results with different initial fiberorientations are depicted in FIG. 191. The draw-in distance comparisonis listed in Table 9-20. It can be seen that most of the draw-indistances predicted by simulation are within the ranges of theexperiment results. The two largest deviations happened, however, in thex direction draw-in for −45/+45 initial fiber orientation in both singlelayer and double layers cases. It is also worth noting that for singlelayer −45/+45 setup, in experiments, specimen size in the x directionactually became larger (negative draw-in distance) after preforming,which is different from not only the simulation, but also common formingprocesses.

TABLE 9-20 Double-dome draw-in distance comparison. Initial exp ×draw-in/ sim × draw-in/ exp × draw-in/ exp × draw-in/ orientation inchinch inch inch  0/90 0.70 about 1.34 1.50 1.27 about 1.71 1.73 −45/+45−0.60 about −0.15 0.10 1.43 about 1.95 1.70 0/90 in 1.00 about 1.38 1.001.53 about 1.61 1.46 0/90/−45/+45 −45/+45 in 0.25 about 0.32 1.45 0.99about 1.19 1.62 0/90/−45/+45

For warp and weft yarn angle distribution, results are shown in FIG.192. It can be seen that for the single layer cases, the simulationagrees well with the experiment. At nearly all measuring locationssimulation results are within experimental deviation and achieve theproposed 5% prediction error compared to the averaged experimentalresults. For the double layer, however, the discrepancy is larger. Themost possible reason for this discrepancy is that the rapid drop ofprepreg temperature, as shown in panel (b) of FIG. 190, results in resinmelting and re-solidify between two prepreg layers, which causes verytacky or even disappeared prepreg interface, leading to much largerinteraction strength and makes relative sliding between prepregs a lotmore difficult compared to simulation. Actually, from an experimentalaspect, the supplied prepreg in the invention has the best preformingtemperature at 50 to 80° C. when the resin is molten but not cured. Whenthe temperature drops down to around 23° C., the resin will be hard andvery sticky, leading to undesired features such as edge breakage,discontinuous deformation, and out-of-plane deformation or even folding,as shown in FIG. 193. In the future, not only the temperature-dependentprepreg surface interaction should be taken into account in preformingsimulation, but also it might be necessary to adjust the temperature ofthe tools by warming the coolant to the desired temperature. The purposefor this operation is not simply to reduce simulation and experimentdiscrepancies, but more importantly, it can improve the quality of thefinal parts.

The double-dome preforming benchmark test established can introducecomplex double curvature features at the size of common automobileparts. Combined with the corresponding quantitative measurement of localprepreg temperature history, draw-in distance, local yarn angle, andforming force also developed in the invention, it serves as an effectiveexperimental approach to validate the preforming simulation methodsdeveloped in the invention. Validation results indicate that thedeveloped models can reach the proposed fiber orientation predictionerror of less than 5% most of the time, guaranteeing the models'application potential.

Besides for calibrating the models in the invention, this double-domebenchmark test can also serve as validation for preforming simulationmodels developed in the future or from other researchers due to the factthat it considers most of the process parameters and provides the mostimportant criteria for the final parts' performance. As a result, thisapproach enables researchers in both academic and industrial fields totest their preforming models in a reliable way, so it motivates theinvention of accurate preforming simulation models that can helpincrease production and broaden application of advanced CFRPs, whilebenefiting environmental emission and fossil fuel control.

Moreover, the double-dome preforming tests performed in the inventionprovide important information about the production of high-qualityparts. Temperature control in not only the heated oven but also withforming tools is essential for the resin to fully melt and cause smallprepreg deformation resistance and small prepreg surface interaction,which are the keys to smooth and defect-free final parts.

Despite the fact that this exact setup of the double-dome preformingvalidation experiment is difficult to be commercialized directly, itfacilitates commercialization of the preforming models developed in theinvention for validation of these models' prediction accuracy. Thequantitative measurement approaches for local temperature, draw-indistance, local yarn angle, and forming force can be transferred toother research teams in an open source form to establish awidely-accepted preforming simulation validation standard, which canaccelerate development of other high-accuracy preforming models.Temperature control and monitor experience gained from this preformingexperiment can be implemented into preforming presses for realproduction in the form of heated coolant and embedded thermocouples, toproduce smooth and defect-free CFRP parts. In the part-level preformingmodel validation according to the invention, we established adouble-dome benchmark test and corresponding quantitative measurementapproaches and criteria. The benchmark tests validate the predictioncapabilities of the preforming models we developed. Moreover, itprovides a trustworthy approach to test models from other researchers,and gives insight guidance for the design of preforming facilities.

For experiments with multiple parameters that need to be considered,like this preforming benchmark test, it is essential to properly performthe design of the experiment to clearly study the effects fromparameters, while keeping material and time consumption low. Forpreforming using prepregs, it is important to control not only initialprepreg temperature, but also tool temperature to ensure the resin keepsmelting during the whole preforming processes for smooth and defect-freeparts production.

The foregoing description of the exemplary embodiments of the presentinvention has been presented only for the purposes of illustration anddescription and is not intended to be exhaustive or to limit theinvention to the precise forms disclosed. Many modifications andvariations are possible in light of the above teaching.

The embodiments were chosen and described in order to explain theprinciples of the invention and their practical application so as toactivate others skilled in the art to utilize the invention and variousembodiments and with various modifications as are suited to theparticular use contemplated. Alternative embodiments will becomeapparent to those skilled in the art to which the present inventionpertains without departing from its spirit and scope. Accordingly, thescope of the present invention is defined by the appended claims ratherthan the foregoing description and the exemplary embodiments describedtherein.

1. A method for design optimization and/or performance prediction of amaterial system, comprising: generating a representation of the materialsystem at a number of scales, wherein the representation at a scalecomprises microstructure volume elements (MVE) that are of buildingblocks of the material system at said scale; collecting data of responsefields of the MVE computed from a material model of the material systemover predefined sets of material properties/constituents and boundaryconditions; applying machine learning to the collected data of responsefields to generate clusters that minimize a distance between points in anominal response space within each cluster; computing an interactiontensor of interactions of each cluster with each of the other clusters;manipulanting the governing partial differential equation (PDE) usingGreen's function to form a generalized Lippmann-Schwinger integralequation; and solving the integral equation using the generated clustersand the computed interactions to result in a response prediction that isusable for the design optimization and/or performance prediction of thematerial system.
 2. The method of claim 1, further comprising passingthe resulted response prediction to a next coarser scale as an overallresponse of that building block, and iterating the process until a finalscale is reached.
 3. The method of claim 1, wherein the building blocksare defined by material properties and structural descriptors obtainedby modeling or experimental observations and encoded in a domaindecomposition of structures for identifying locations and properties ofeach phase within the building blocks.
 4. The method of claim 3, whereinthe structural descriptors comprise characteristic length and geometry.5. The method of claim 1, wherein the boundary conditions are chosen tosatisfy the Hill-Mandel condition.
 6. The method of claim 1, wherein thecollected data of response fields comprise a strain concentrationtensor, a deformation concentration tensor, stress tensor including PK-Istress and/or Cauchy stress tensors, plastic strain tensor, thermalgradient, or the like.
 7. The method of claim 1, wherein the machinelearning comprises unsupervised machine learning and/or supervisedmachine learning.
 8. The method of claim 1, wherein the machine learningis performed with a self-organizing mapping (SOM) method, a k-meansclustering method, or the like.
 9. The method of claim 1, wherein theclusters are generated by marking all material points that have the sameresponse field within the representation of the material system with aunique ID and grouping material points with the same ID.
 10. The methodof claim 9, wherein the generated clusters is a reduced representationof the material system, which reduces the number of degrees of freedomrequired to represent the material system.
 11. The method of claim 10,wherein the generated clusters are a reduced order MVE of the materialsystem.
 12. The method of claim 1, wherein the computed interactiontensor is for all pairs of the clusters.
 13. The method of claim 1,wherein said computing the interaction tensor is performed with fastFourier transform (FFT), numerical integration, or finite element method(FEM).
 14. The method of claim 1, wherein the PDE is reformulated as aLippmann-Schwinger (LS) equation.
 15. The method of claim 14, whereinsaid solving the PDE with the LS equation is performed with arbitraryboundary conditions and material properties.
 16. The method of claim 1,wherein the collected data of response fields, the generated clusters,and/or the computed interaction tensor are saved in one or more materialsystem databases.
 17. The method of claim 16, wherein said solving thePDE with the LS equation is performed in real-time by accessing the oneor more material system databases for the generated clusters and thecomputed interaction tensors.
 18. A method for design optimizationand/or performance prediction of a material system, comprising:performing an offline data compression, wherein original microstructurevolume elements (MVE) of building blocks of the material system arecompressed into clusters, and an interaction tensor of interactions ofeach cluster with each of the other clusters is computed; andmanipulanting the governing partial differential equation (PDE) usingGreen's function to form a generalized Lippmann-Schwinger integralequation; and solving the integral equation using the generated clustersand the computed interactions to result in a response prediction that isusable for the design optimization and/or performance prediction of thematerial system.
 19. The method of claim 18, further comprising passingthe resulting response prediction to a next coarser scale as an overallresponse of that building block, and iterating the process until a finalscale is reached.
 20. The method of claim 18, wherein the buildingblocks are defined by material properties and structural descriptorsobtained by modeling or experimental observations and encoded in adomain decomposition of structures for identifying locations andproperties of each phase within the building blocks.
 21. The method ofclaim 20, wherein the structural descriptors comprise characteristiclength and geometry.
 22. The method of claim 18, wherein the boundaryconditions are chosen to satisfy the Hill-Mandel condition.
 23. Themethod of claim 18, wherein said performing the offline data compressioncomprises: collecting data of response fields of the MVE computed from amaterial model of the material system over a predefined set of materialproperties and boundary conditions; applying machine learning to thecollected data of response fields to generate clusters that minimize adistance between points in a nominal response space within each cluster;and computing the interaction tensor is for all pairs of the clusters.24. The method of claim 23, wherein the collected data of responsefields comprise a strain concentration tensor, a deformationconcentration tensor, stress tensor including PK-I stress and/or Cauchystress tensors, plastic strain tensor, thermal gradient, or the like.25. The method of claim 23, wherein the machine learning comprisesunsupervised machine learning and/or supervised machine learning. 26.The method of claim 23, wherein the machine learning is performed with aself-organizing mapping (SOM) method, a k-means clustering method, orthe like.
 27. The method of claim 23, wherein the clusters are generatedby marking all material points having the same response field within therepresentation of the material system with a unique ID and groupingmaterial points with the same ID.
 28. The method of claim 27, whereinthe clusters is a reduced representation of the material system, whichreduces the number of degrees of freedom required to represent thematerial system.
 29. The method of claim 28, wherein the clusters areadapted as a reduced order MVE of the material system.
 30. The method ofclaim 23, wherein said computing the interaction tensor is performedwith fast Fourier transform (FFT), numerical integration, or finiteelement method (FEM).
 31. The method of claim 23, wherein the PDE isreformulated as a Lippmann-Schwinger (LS) equation.
 32. The method ofclaim 31, wherein said solving the PDE with the LS equation is performedwith arbitrary boundary conditions and material properties.
 33. Themethod of claim 23, wherein the collected data of response fields, thegenerated clusters, and/or the computed interaction tensor are saved inone or more material system databases.
 34. The method of claim 33,wherein said solving the PDE with the LS equation is performed withonline accessing the one or more material system databases for thegenerated clusters and the computed interactions.
 35. A non-transitorytangible computer-readable medium storing instructions which, whenexecuted by one or more processors, cause a system to perform a methodfor design optimization and/or performance prediction of a materialsystem, wherein the method is in accordance with claim
 1. 36. Acomputational system for design optimization and/or performanceprediction of a material system, comprising one or more computingdevices comprising one or more processors; and a non-transitory tangiblecomputer-readable medium storing instructions which, when executed bythe one or more processors, cause the one or more computing devices toperform a method for design optimization and/or performance predictionof a material system, wherein the method is in accordance with claim 1.37. A material system database usable for conducting efficient andaccurate multiscale modeling of a material system, comprising: clustersfor a plurality of material systems, each of which groups all materialpoints having a same response field within a region within amicrostructural volume element (MVE) of a respective material systemwith a unique ID; interaction tensors, each of which representsinteractions of all pairs of the clusters (regions with unique ID) forthe respective material system; and response predictions computed basedon the clusters and the interaction tensors.
 38. The material systemdatabase of claim 37, wherein the clusters are generated by applyingmachine learning to data of response fields of the MVE computed from amaterial model of the respective material system over a predefined setof material properties and boundary conditions.
 39. The material systemdatabase of claim 38, wherein the interaction tensors are computed withfast Fourier transform (FFT), numerical integration, or finite elementmethod (FEM).
 40. The material system database of claim 39, wherein theresponses predictions are obtained by solving a governing partialdifferential equation (PDE) with the LS equation using the clusters andthe computed interactions.
 41. The material system database of claim 40,wherein the responses predictions comprise at least stiffness, stressresponses, damage initiation, fatigue indicating parameter (FIP), and/orthermal expansion.
 42. The material system database of claim 37, beingconfigured such that some of the response predictions are assigned as atraining set for training a different machine learning that connectsprocesses/structures to responses/properties of the material systemdirectly without going through the clustering and interaction computingprocesses at all; and some or all of the remaining response predictionsare assigned as a validation set for validating the efficiency andaccuracy of the multiscale modeling of the material system.
 43. Thematerial system database of claim 37, being generated with predictivereduced order models.
 44. The material system database of claim 42,wherein the predictive reduced order models comprise a self-consistentclustering analysis (SCA) model, a virtual clustering analysis (VCA)model, and/or an FEM clustering analysis (FCA) model.
 45. The materialsystem database of claim 37, being updatable, editable, accessible, andsearchable.
 46. A method of applying the material system database ofclaim 37 for design optimization and/or performance prediction of amaterial system, comprising: training a neural network with data of thematerial system database; and predicting real-time responses of thematerial system during a loading process performed using the trainedneueral network, wherein the real-time responses are used for the designoptimization and/or performance prediction of a material system.
 47. Themethod of claim 46, further comprising performing a topologyoptimization to design a structure with microstructure information. 48.The method of claim 46, wherein the neural network comprises a feedforward neural network (FFNN) and/or a convolutional neural network(CNN).
 49. A non-transitory tangible computer-readable medium storinginstructions which, when executed by one or more processors, cause asystem to perform a method for design optimization and/or performanceprediction of a material system, wherein the method is in accordancewith claim 46.